x between a and b, the author tries what ought to be the value of tan 00 , if this principle be true, and finds + V—1, which on trial is found to satisfy the fundamental equations of trigonometry. Section 4. — Series of alternately positive and negative signs stand upon a much safer basis than those in which all the terms have the same signs, and that whether their divergence be finite or infinite. It has long been observed, that when the terms of an alternating series begin by diminishing, even though they afterwards increase, the converging portion may be made effective in approximating to the arithmetical equivalent of the series. The error committed by stopping at any term is not so great as the first of the rejected terms. In many alternating series this has been proved to be true, and it seems never to have been supposed that the theorem was anything but universal. In this section instances are produced in which the theorem is not true ; and at the same time various proofs of it are given, each of which applies to very extensive cases, and the tendency of which is to show that it is only under definite and unusual condi- tions that the theorem can fail. Still, however, no positive criterion is established for ascertaining whether the theorem be true or not in any particular case. Section 5. — On double infinite series, in which the terms are in- finitely continued in both directions. It seems, in many different ways, that the series . . . + 0(j?-2)-l-^(j?-l) + x +

m -J— m the first, and making similar changes in the other two.
The particular conditions which must be satisfied at the boundaries
18
of the fluid are then considered, and the general equations applied
to a few simple cases.
On considering these equations the author was led to observe,
that both Lagrange's and Poisson's proofs of the theorem that udx
+ vdy + wd2 is always an exact differential when it is so at any
instant (the pressure being supposed equal in all directions), would
still apply, whereas the theorem is manifestly untrue when the tan-
gential force is taken into account. This led him to perceive that
one objection to these proofs is of essential importance. He has
given a new proof of the theorem, which however was not necessary
to establish it, as it has been proved by M. Cauchy in a manner
perfectly satisfactory.
The methods employed in this paper in the case of fluids apply
with equal facility to the determination of the equations of equili-
brium and motion of homogeneous, uncrystallized, elastic solids, the
only difference being that we have to deal with relative velocities in
the former case, and with relative displacements in the latter. The
only assumption which it is necessary to make, is that the pressures
are linear functions of the displacements, or rather relative displace-
ments, the displacements being throughout supposed extremely small.
The equations thus arrived at contain two arbitrary constants, and
agree with those obtained in a different manner by M. Cauchy. If
we suppose a certain relation to hold good between these constants,
the equations reduce themselves to Poisson's, which contain but one
arbitrary constant.
The equations of fluid motion which would have been arrived at
by the method of this paper if the two constants t, f had been re-
tained, have been already obtained by Poisson in a very different
manner. The author has shown, that according to Poisson's own
principles, a relation may be obtained between his two constants,
which reduces his equations to those finally adopted in this paper.
There is one hypothesis made by Poisson in his theory of elastic
solids, by virtue of which his equations contain but one arbitrary
constant, which the author has pointed out reasons for regarding as
improbable. He has also shown that there is ground to believe that
the cubical compressibility of solids, as deduced by means of Pois-
son's theory from their extensibility when formed into rods or wires,
is much too great, a conclusion which he afterwards found had been
previously established by the experiments of Prof. Oersted.
The equations of motion of elastic solids with two arbitrary con-
stants, are the same as those which have been obtained by different
authors as the equations of motion of the luminiferous sether in va-
cuum. In the concluding part of his paper the author has endea-
voured to show that it is probable, or at least quite conceivable, that
the same equations should apply to the motion of a solid, and to
those.very small motions of a fluid, such as the Kther, which accord-
ing to the undulatory theory constitute light.
19
May 12. 1845.
On the Aberration of Light. By G. G. Stokes, M.A,, Fellow of
Pembroke College.
In the common explanation of aberration, it is supposed that light
comes in a straight line from a heavenly body to the surface of the
earth, except in so far as it is bent by refraction. This, of course,
would follow at once from the theory of emissions ; but it appears
at first sight difficult to reconcile with the theory of undulations,
unless we make the startling supposition that the aether passes
freely through the earth as the earth moves round the sun. The
object of this paper is to show that if we make the following suppo-
sitions, that the earth in its motion pushes the aether out of its way,
that the aether close to the surface of the earth is at rest relatively
to the earth, and that light is propagated through the disturbed
aether as we suppose sound to be propagated air in motion, the ob-
served law of aberration wull still result, provided the motion of the
aether be such that udx + vdy + wdz is an exact differential, where
u, V, w are the resolved parts of the velocity of any particle of the
se her along the rectangular axes of x, y, z.
On the Pure Science of Magnitude and Direction. By the Rev.
H. Goodwin, Fellow of Caius College, and of the Cambridge Philo-
sophical Societ)'.
This memoir may be considered in some degree supplementary to
the preceding one by the same author, " On the Connexion of the
Sciences of Mechanics and Geometry." In that memoir it was
argued, that if the views there advanced were sound, there must be
such a science as that of pure direction, or rather a pure science of
magnitude and direction which should include within itself the sciences
of geometry, of kinematics, and of mechanics ; in this the attempt is
made to establish mathematically the fundamental proposition of
such a science.
By making use of De Moivre's formula, the author conceives him-
self to have established this proposition, that if P represents the
magnitude of any cause which varies uniformly and continuously
into its exact opposite, i. e. into — P while its direction varies uni-
formly from a given direction to the exactly opposite direction ; and
if be the angle which the direction of P makes Asath a given direc-
tion, then P is equivalent to two causes, P cos fl in that given direc-
tion, and P sin 9 in the direction perpendicular to it.
The author is aware of the hnprobability which may appear to
exist, that so general a proposition should be susceptible of proof
without reference to particular instances, and has therefore endea-
voured to obviate some objections, which will be more or less
strongly felt, according to the nature of the philosophy of knowledge
adopted by the mind which makes them, and which in some cases
will probably be invincible.
The memoir concludes with some remarks on the general question
20
of the transition of a quantit)'^ from the + to the — afFection, which
the author conceives to be illustrative of his general design.
December 8, 1845.
On the Heights of the Aurora Borealis of September 17 and
October 12, 1833. By Professor Potter, A.M., of Queen's College.
The data for the calculations are almost entirely taken from the
consjjectuses of the observations printed and distributed in 1833 by
the British Association ; and although so long time has elapsed, no
calculations of the heights of the phsenomena, which are the first
steps to be taken in finding the nature of the meteor, have, to the
author's knowledge, been hitherto published; the only imperfect
discussion being given in the Philosophical Magazine for December
1833.
The observers of the display of September 1 7, were Mr. J. Phil-
lips, at York ; Mr, Clare, Mr. Hadfield and the author, at or near
Manchester ; Professor Airy, at Cambridge ; and the Hon. C. Harris,
near Gosport.
The observers of that on October 12, were Professor Sedgwick, at
Dent; Mr. W. L. Wharton, near Guisborough ; Mr. J. Phillips, at
York ; Mr. Clare, Mr. Hadfield and the author, at or near Man-
chester ; Dr. Robinson, at Armagh ; Professor Airy, at Cambridge ;
and the Hon. C. Harris, at Heron Court.
The observations of the aurora of September 17 at Cambridge at
8^ 25"™ Greenwich time, taken with those at Manchester at 8^ 24™,
give the height of the lower edge of the arch 56 English miles, and
of „ equal to m
terms of the following series : —
A„_i+1.A„_2 + 1.2.A„_3 + 1.2.3A„.4+ &c. +1.2... («-i+ 1) A.-
+ &C. +1.2...(«— l)Ao
4-A„_i + 2.A„_2+2.3A„_3 + 2.3.4A„_4+ &c. +2.3 ...(n-i+2) A^
+ &c. +2.3. ..wAo
+A._i + 3.A„_2+3.4A,._3 + 3.4.5 A„_,+ &c. +3.4...(w-i+3) A.
+ &C. +3.4...(»+l)Ao
+ A,.i + 4.A„_2+4.5A„_3 + 4.5.6 A„_4+ &c. +4.5 ...(«-i+4) A^
+ &c. +4.5...(« + 2)Ao
&c. &c.
where
Ai=(n—i)(pi—hiPi_^ + h2Pi.i+&.c. + {—iyKpi.r + &c.±hi.iPi + hi)
and
Ai=the sum of the natural numbers 1, 2, 3 (n—i).
A2=the sum of the homogeneous products of the same quantities of
two dimensions.
Agssthe sum of the homogeneous products of the same quantities of
three dimensions, and so forth.
From the above formula for A^ may be determined all the coeffi-
cients A except the first, which is determined from the equation
A„_i=i?„-i— i?„-2+iJn-3— &c. +;?!+ 1.
Having determined the quantities A in any particular case, let them
he substituted in the first line of the series. If the sum of that line
be equal to —p„ unity is a root of the equation. Let the second line
be then written down and added to the first. If the sum of the two
equals— jt7„, 2 is a root of the equation, and so by adding successive
lines we shall ascertain whether the successive integers 3.4., &c. are
or are not roots of the equation.
The quantities h in the expression for A; depend upon the number
of the coefficient and the number of the dimensions of the equation.
The author proposes that these should be calculated and tabulated
for equations of all dimensions up to a certain limit, by which means
we should be in possession of so many skeletons of equations, ready
for application in any particular case, and the calculation in particular
instances would be thus greatly facilitated.
It will be observed that each successive line is derived from that
preceding by a simple division and multiplication of the separate
terms of the latter, and thus each succeeding trial facilitates those
which follow ; contrary to what obtains in the ordinary method by
successive substitutions, in which each attempt proceeds de novo.
If the addition of a term makes the series from being greater than
p„ less than it, or vice versd, a fractional or surd root will lie between
81
the number expressing the number of the term so added and the
number next below it.
If all the roots are impossible, the series will be either always
greater or always less than p^, whatever be the number of terms
taken.
For an example take the cubic
s^ -\-px^ -\-qx-\-r=^0.
Here — r=3 x 1.2 + 2(j)— 3)l+y-j9+ 1
+ 3x2.3 + 2(jo-3)2 + y-;;+l
+ 3x3.4 + 2(;j— 3)3 + ?— /?+l
4- &c. to X terms,
if a; is a positive integer.
The method in common with other experimental methods applies
to the discovery of all roots, possible or impossible, which do not
involve surds.
March 12, 1849.
On the Intrinsic Equation to a Curve, and its application. By
the Master of Trinity.
The author remarked that the expressions for the lengths of
curves, their involutes and evolutes, in the ordinary methods, are
complex and untractable, which arises in a great measure from the
properties of extrinsic lines being introduced, namely, coordinates.
But a curve may be represented without any such additions, by an
equation Ijetween the length and the angle of flexure, which is
therefore called the intrinsic equation. This equation gives, with
remarkable facility, the radii of curvature ; involutes and evolutes
of most curves. It also expresses very simply what may be called
running curves ; namely, curves which run like a pattern along a
strip of ornamented work. A very simple equation expresses, for
instance, the inclined scroll pattern so common in the antique, and
by altering the constants, gives to this pattern an endless variety of
forms. If s be the length of the curve and (p the angle, the intrinsic
equation to the circle is 5 = a^; to the cycloid s= a sin (p. The
equation to an epicycloid or hypocycloid is s = a sin 7«^, according
as m is less or greater than unity. The equation to an undulating
pattern is

)•) No Xs are some Ys
O, (•( Some Xs are no Ys.
Previously to entering upon the forms of syllogism, Mr. De Morgan
repeats and reinforces the objections brought forward in his Formal
Logic; namely, that )( is a compound of )) and ((, and has no sim-
ple contradiction in the system ; and that (•) not only has no simple
contradiction, but cannot be contradicted except when the terms are
singular and identical. He then proceeds to propose one mode of
remedying these defects. Calling the ordinary proposition cumular,
he proposes to make it exemplar, as asserting or denying of one in-
stance only. In the universal proposition, the example is wholly in-
definite, any one ; in the particular proposition it is not wholly indefinite,
some one. The defects of contradiction are thus entirely removed, as
in the following list, in which each universal proposition is followed
by its contradiction.
93
)( Any one X is any* one Y
(•) Some one X is not some one Y
)) Any one X is some one Y
(•( Some one X is not any one Y
(( Some one X is any one Y
)•) Any one X is not some one Y
) • ( Any one X is not any one Y
Some one X is some one Y
In both systems there are thirty-six valid syllogisms, and in both
the canon of validity is, — one universal (or wholly indefinite) middle
term, and one affirmative proposition. But the symbolic canons of
inference diflFer as follows (with reference to the order XY, YZ, XZ).
Exemplar system. — Erase the middle parentheses, and the symbol
of the conclusion is left : thus ())•) gives (•).
Cumular system. — Erase the middle parentheses, and then, if both
the erased parentheses turn the same way, turn any universal paren-
thesis which turns the other way, unless it be protected by a mark
of negation. Thus )•(() gives )•), ())( gives (), and ())•( gives (•(.
Section V. On the theory of the copula, and its connexion with
the doctrine of figure. — In his work on Formal Logic, Mr. De Morgan
had analysed the copula, and abstracted what he calls the copular
conditions of the relation connecting subject and predicate. These
are, transitiveness, seen in such copulse as support, govern, is greater
than, &c., ex. gr. if A govern B, and B govern C, A governs C : and
convertibility , seen in such copulse as is acquainted with, agrees with, &c. ;
ex. gr. if A agree with B, B agrees with A. Mr. De Morgan's position
is, that any mode of relation which satisfies both these conditions has
as much claim to be the copula as the usual one, is, which derives its
fitness entirely from satisf}-ing the above conditions. So far the
work cited. In the present paper the correlative copula is introduced,
as is supported in opposition to supports, &c., and every system of
syllogism is thus extended. If a copula be taken which is only trans-
itive, but not convertible, every syllogism remains valid, provided
that the correlative of that copula be used instead of it, when needful.
And in this consists, according to Mr. De Morgan, the root of the
doctrine oi figure. If + represent affirmative, and — negative, the
four figures are connected with + + , H , f- . and (in the
system of contraries, where negative premises may have a VcJid con-
clusion, the fourth figure has equal claims with the rest, though the
conditions of all the figures are singularly altered). These forms
do not require the correlative copula : thus -^ in the second figure
(as Camestres and Baroko among the Aristotelian forms) are as valid
when the copula is 'supports' or 'is greater than,' as when 'is' is
employed. But in every other case the rule for the proper intro-
duction of the correlative copula is as follows : — The preceding being
called the primitive forrns of the four figures, when one premise of a
primitive form is altered, the necessity of a correlative copula is
thrown upon the other ; when both, upon the conclusion. Thus, the
primitive form of the second figure being -\ , and Cesare showing
h. it is only valid with the copula 'governs,' by making ' is not
governed by ' the copula of the conclusion, as follows : —
No Z governs any Y
Every X governs a Y
Therefore no X t5 governed by any Z.
* So that there can be but one X and one Y, and that X is Y.
94
By an additional letter (ff) introduced into the usual words of
syllogism, the places of the correlative copula may be remembered,
as in Barbara, Celagrent, &c. : a g being made to accompany any
member of the syllogism in which the correlative copula must be
employed.
This theory is applied equally to the Aristotelian system, to Sir
William Hamilton's (though not of universal application in the cu-
mular form), and to Mr, De Morgan's system of contraries. The
extensions required by the use of a merely transitive copula, in the
last-mentioned system, are discussed ; and mention is made of the
tricopular system, in which the leading copula and its correlative
have an intermediate or middle relation, equally connected with
both ; as in > = and < of the mathematicians.
The next step is the assertion that it is not necessary that any
two of the three copulse of a syllogism should be the same ; all that
is requisite is that, in affirmative syllogisms, the copular relation in
the conclusion should be compounded of those in the premises. The
instrumental part of inference is described by Mr. De Morgan as the
elimination of a term hy composition (including resolution) of relations,
which leads to the conclusion thdX whenever a negative premise occurs,
there is a resolution of a compound relation. This resolution is shown
in a case (among others) of the ordinary copula, in which, however,
it would hardly strike the mind more forcibly than would the pro-
perties of powers in algebra if every letter represented unity. Mr.
De Morgan shows (in an addition) that in some isolated cases of in-
ference which are not reducible to ordinary syllogism, logicians have
had recourse to what amounts to composition of relations.
Mr. De Morgan next points out that the copular relation, in
affirmative propositions, need not be restricted as applying to one
instance only of the predicate ; and shows that the removal of this
usual restriction entirely removes all his objections to Sir William
Hamilton's form of his own system.
Section VI. On the application of the theory of probabilities to some
questions of evidence. — This inquiry was suggested by the apparent
(but only apparent) error of the logicians, who seem to lean towards
the maxim that, when the subject and predicate are unknown, the
universal and particular propositions ' Every X is Y,' ' Some Xs are
not Ys,' are a priori of equal probability. The difficulty is one which
occurs in the following case : — If a good witness, drawing a card
from a pack, were to announce the seven of spades, his credit would
not be lowered, though he would have asserted an event against which
it was 51 to 1 (i priori. A common person gives the true answer,
' Why not the seven of spades as well as any other ? ' Many readers
of works on probability would be inclined to say ' That is not the
question ; why the seven of spades rather than some one or another
of the fifty- one others ? ' The retort is fallacious : it rubs out the
distinctive marks from the other fifty-one cards, and writes on each
of them ' not the seven of spades ' as its only exponent. Laplace
has chosen two problems, in one of which the distinctive marks
exist, and not in the other ; and, neglecting the consideration of the
first one, has founded his remarks upon the deterioration of evidence
95
by the assertion of an improbable event, entirely upon the second,
'fhe object of this section is, by a closer examination of the mathe-
matical problem of evidence, to ascertain the accordance or non-
accordance of the results of usual data with usual notions. The
result of the examination is, that common notions, as in other cases,
are found closely accordant with theory. For instance, if there be
n possible things which can happen, so that the mean probability of
an event is -, a witness of whom we know no particular bias towards
n
one mode of error rather than another, asserting an event of which
the d, priori probability is a, has his previous credit raised, unaltered,
or lowered, according as a— — is positive, nothing, or negative. So
n
that though the d, priori probabilities were distributed among a mil-
lion of possible and distinguishable cases, yet a witness asserting one
of them against which it is only 999,999 to 1, would have as good
a right to be beheved as though there had been but two equally pro-
bable cases, of which he had asserted one.
March 11, 1850.
On the Numerical Calculation of a class of Definite Integrals and
Infinite Series. By Professor Stokes.
In a paper " On the Intensity of Light in the neighbourhood of a
Caustic," printed in the sixth volume of the Cambridge Philosophical
Transactions, Mr. Airy, the Astronomer Royal, has been led to con-
sider the integral
W= / COS — (w^ — mw)dw,
Jo 2^ ^ '
and has tabulated it from m=—4 to m=+4 by the method of qua-
dratures. In a supplement to the same paper, printed in the fifth
part of the eighth volume, Mr. Airy has extended the table as far as
m= + 5"6, by means of a series proceeding according to ascending
powers of m. This series, though convergent for all values of m,
however great, is extremely inconvenient for numerical calculation
when m is large, and moreover gives no information as to the law of
the progress of the function for large values of m. The author has
obtained the following expression for the calculation of W for large,
or even moderately large, positive values of m :
where
W=2(3m)-i|Rcos { (y'. XY'-Y, X, J,. -^,...)=o
can be solved ; it is seen that the original equation can be solved by
eliminating X between x=Y' and y=XY'—Y.
'i'he general method of which this is a particular case, is as follows.
Letf(x, y, X, Y)=0 have its differential equations of the first order
formed on two suppositions : first, that X and Y aire constant ; se-
condly, that X and y are constant. Let these differential equations
be
X=*(^, y, y<) X = cp(X, Y, Y')
Y=^(x, y. y') y=J.(X, Y, Y').
These equations may be used instead of the first two pairs of cor-
relatives in the preceding example : and each differential coeflficient
of Y is expressible by means of the same and lower differential
coefficients of y ; and vice versd. To get convertible forms, as in
the instance above, /(j?, y, X, Y) must be chosen so that .r and y are
simultaneously interchangeable with X and Y.
Mr. De Morgan gives a similar extension of the method as applied
to partial differential equations.
3- On the constants of a primitive equation. — It is usually left to
109
be collected from induction that the equation of the rath order has n
constants, and no more, in its complete primitive. Mr. De Morgan
proposes an (I priori proof of this point, on which, as in all such
cases, it would be presumptuous to decide until it has been tho-
roughly examined.
He further proposes an extension of the meaning of the term so-
lution, in the case of all the primitives intermediate between the
differential equation and the original primitive. Thus, supposing
an equation of the third order, of which the admitted primitives of
the second order are
Uj= const., 1/2= const., 1/3= const.,
he maintains that the general primitive of the second order is
/(U„ U^, U3)=0.
where/ is any function whatsoever: and, starting from this last
equation, he determines a general primitive of the first order in a
similar way.
This view is supported by the reduction of a common differential
equation of the nth order to a partial differential equation of the
first order with n independent variables.
4. On the criterion of integrahility of (p{x, y, y', y",...). — If
we denote the differential coefficients of y hyp, q, r, s, &c., it is well
known that the condition which is both necessary and sufficient, in
order that V=^(cr, y, p, ?....) may be integrable without reference
to relation between y and x, is
v,-v;+v,"-v;"+...=o,
the accent denoting complete differentiation with respect to x. This
has usually been established, either by the calculus of variations, or
by a process of elaborate expression of the actual result in terms of
definite integration with respect to a subsidiary variable. Mr. De
Morgan, after some remarks upon the manner in which certain
proofs of the necessity of the criterion fail, gives a very simple ele-
mentary proof founded upon the following theorem. If U be any
function of x, y, p, &c., — as far say as s, for an instance, — then
(u').=u;, (u%=u;. (u%=u;+u,.
(u'),=u,'+u„ (u').=u/+u„ (u').=u;+u,. (U').=u,.
Mr. De Morgan takes it to have been hitherto unnoticed that the
formulae V^— V,'4-V,"— ..., V,— V/ + .... so much used in this sub-
ject, are, when V is integrable, nothing but the differential coefficients
of fVdz, with respect to y,p, &c.
[But since the paper was communicated, Mr. De Morgan has found
the above theorem, and its consequences, in a memoir by M. Sarrus,
apparently belonging to the Journal de VEcole Poly technique, and
printed in 1824. No notice is taken of this method by MM. Ber-
trand, Binet, or Moigno, who have written on the subject since
M. Sarrus.] .
110
May 5, 1851.
Of the Transformation of Hypotheses in the History of Science.
By W. Whewell, D.D.
The author remarks that new theories supersede old ones, not
only by the succession of generations of men, but also by transfor-
mations which the previous theories undergo. Thus the Cartesian
hypothesis of vortices was modified so that it explained, or was sup-
posed to explain, a central force : and then, the Cartesian philoso-
phers tried to accommodate this explanation of a central force to the
phsenomena which the Newtonian principles explained ; so that in
the end, their theory professed to do all that the Newtonian one did.
The machinery of vortices was, however, a bad contrivance to pro-
duce a central force ; and when it was applied to a globe, its defect
became glaring. Still however, the doctrine of vortices has in it
nothing which is absurd anterior to observation. The " nebular
hypothesis" is a hypothesis of vortices with regard to the origin of
the system of the universe, and is now held by eminent philosophers.
Nor is the doctrine of the universal gravitation of matter at all in-
consistent with some mechanical explanation of such a property ;
for instance, Le Sage's. We cannot say therefore that if the planets
are moved by gravitation, they are not moved by vortices. The Carte-
sians held that they were moved by both ; by the one, because by
the other.
Like remarks may be made with respect to the theories of mag-
netism and of light.
May 19, 1851.
On the Colours of Thick Plates. By G. G. Stokes, M.A., Fellow
of Pembroke College, and Lucasian Professor of Mathematics in the
University of Cambridge.
By the expression " colours of thick plates " is usually understood
the system of coloured rings, discovered by Newton, which are formed
on a screen when the sun's light is transmitted through a small hole
in the screen, and received perpendicularly upon a concave mirror of
quicksilvered glass, placed at such a distance from the screen that
the image of the hole is at the same distance from the mirror as the
hole itself. The brilliancy of the rings, as was afterwards discovered,
is greatly increased by tarnishing the surface of the mirror ; and it
is also advantageous to use a lens to collect the sun's rays, and to
place the screen so that the small hole may be situated at the focus
of the lens. These rings were first explained on the undulatory
theory by Dr. Young, who attributed them to the interference of
two streams of light, of which the first is scattered at the tarnished
surface of the mirror, and then regularly reflected and refracted, while
the second is regularly refracted and reflected, and then scattered in
coming out of the glass. The theory has been worked out in detail
Ill
by Sir John Herschel, who has investigated the case in which the
two surfaces of the glass belong to a pair of concentric spheres, and
the hole in the screen is situated in the common centre of curvature.
A set of coloured bands has since been observed by Dr. Whewell
in a common plane mirror. These bands are seen when a candle is
held near the eye, at the distance of several feet from the mirror,
and is viewed by reflexion. It is necessary that the first surface of
the glass should be a little tarnished. The theory of these bands
had not been worked out, and it had even been doubted by some
philosophers whether they were of the nature of the coloured rings
of thick plates.
In this paper the author gave a general investigation, which in-
cludes as particular cases the theory of the rings formed on a screen
in Newton's experiment, and that of the bands which Dr. Whewell
had observed in a plane mirror, and which are not thrown on a
screen, but viewed directly by the eye. He also exhibited to the
meeting a variation of Newton's experiment, in which an extremely
beautiful system of rings is very easily produced without sunlight.
The face of a concave mirror of quicksilvered glass was prepared by
pouring on it a mixtvu-e consisting of one part of milk to three or
four of water, and then holding the mirror vertically in front of a
fire to dry. When the flame of a taper, or of an oil-lamp with a
small wick, is placed in front of a mirror thus prepared, in such a
position as to coincide with its inverted image, a beautiful system of
rings is seen encompassing the flame. These rings appear to have
a definite position in space, like a bodily object. The rings thus
formed, which are evidently of the nature of Newton's coloured
rings of thick plates, may be made to pass in a perfectly continuous
manner into the coloured bands observed by Dr. Whewell.
The author has compared theory and experiment in various par-
ticulars, and has found the agreement perfect. It will be suflicient
to mention here one result of theory, which is of great generality
and of considerable elegance. It applies to the system of rings seen
by reflexion in a mirror, either plane or curved, when a luminous
point is placed anywhere near the axis, and the eye occupies any
other position likewise near the axis. The result is as follows : —
Join the eye with the luminous point, and likewise with its image,
whether it be real or virtual, and find the points in which the join-
ing lines, produced if necessary, cut the mirror. Describe a circle
ha\'ing for diameter the line joining these two points. This circle
will be the middle line of the bright colourless fringe of the order
zero, and on each side of it the colours will be arranged in descend-
ing order.
June 2, 1851.
On a new Elliptic Analyser. By Professor Stokes.
After mentioning some of the inconveniences and inaccuracies
attending the use of a Fresnel's rhomb in the analysis of elliptically-
polarized light, and alluding to some other methods which had been
112
employed for the purpose, the author proceeded to describe a new
instrument which he had invented, and which he exhibited to the
meeting. In the construction of this instrument he had aimed at
being independent of the instrument- maker in all important points
except the graduation. The construction is as follows : —
A brass rim or annulus is mounted so as to stand with its plane
vertical when placed on a table. Within this rim turns a brass gra-
duated disc ; and the angle through which it turns is read off by
means of verniers engraved on the face of the rim, and reading to
tenths of a degree. This disc is pierced at the centre, and carries
on the side turned towards the incident light a retarding plate of
selenite, of such a thickness as to give a difference of retardation in
the oppositely polarized pencils amounting to about a quarter of an
undulation. In front it carries a hollow cylinder, turned on the
lathe along with the disc itself. Round this cylinder there turns a
collar containing a Nicol's prism, and carrying a pair of bevel-edged
verniers, by which the angle may be read off through which the
prism has been turned. Thus the retarding plate moves in azimuth
carrying the prism along with it, and the prism has likewise an in-
dependent motion in azimuth.
In observing, the light is extinguished by a combination of the
two movements, in which case the elliptically-polarized light is con-
verted by the retarding plate into plane polarized, which is then ex-
tinguished by the Nicol's prism. On account of chromatic varia-
tions, the light is not, strictly speaking, extinguished, unless homo-
geneous light be employed, but only reduced to a minimum. There
are two principal positions of the retarding plate and Nicol's prism
in which the light is extinguished, or at least would be extinguished
if the incident light were homogeneous ; and for each principal
position there are four subordinate positions, since either the retard-
ing plate or the Nicol's prism may be reversed by turning it through
1S0°. The mean of the four subordinate positions may be taken
for greater accuracy.
Let R, R' be the readings of the fixed, r, r' those of the moveable
verniers in the two principal {)ositions ; I the index error of the fixed
verniers, that is, the azimuth of the major axis of the ellipse de-
scribed, measured from a plane fixed in the disc ; i the index error
of the moveable verniers, that is, the azimuth of the princi])al plane
of the prism, measured from a fixed plane in the disc ; w the angle
whose tangent is equal to the ratio of the axes of the ellipse de-
scribed ; p the difference of retardation of the oppositely polarized
pencils transmitted through the plate, measured as an angle, at the
rate of 360° to one undulation. Then the unknown quantities I, i,
Tn, and p are given in terms of the known quantities R, R', r, and r'
by the following formulae, which happen to be extremely convenient
for numerical calculation : —
I=i(R'-fR); £=l(r'-fr);
cos2«r=4^5V:zrL; cosp= ^an (r'- r)_
sin(R'-R) ' tan(R'-R).
113
The author stated that he liad already observed with this institi-
ment, and after a little practice had found that it worked in a very
satisfactory manner. When the light of the clouds was reflected
horizontally by a mirror, and modified so as to produce ellipti-
cally-polarized light in which the ratio of the axes was about 3 to 1 ,
it was found that the mean error of single observations amounted
to about a quarter of a degree in the determination of the azimuth of
the major axis, about three or four thousandths in the determination
of the ratio of the minor to the major axis, and little more than
the thousandth part of an undulation in the determination of 'p.
Since the magnitude of p depends upon the length of wave, or,
what comes to the same, the refrangibility of the light, it follows
that a knowledge of the former leads to a knowledge of the latter.
It may thus be said that the instrument determines the azimuth and
excentricity of the ellipse described, and the refrangibility of the
light. An error of the thousandth part of an undulation in the de-
termination of p would correspond to an error in the place in the
spectrum assigned to the light operated on amounting to less than
the twentieth part of the interval between the fixed lines D and E.
Now by the use of absorbing media it is possible, without too much
reducing the intensity of the light employed, to alter greatly its mean
refrangibility ; and yet for each medium the refrangibility may be
determined very accurately by means of the value of p. Accord-
ingly, the instrument is specially adapted for investigations relating
to the dispersion of metals, and for other similar researches.
PROCEEDINGS
CAMBRIDGE PHILOSOPHICAL SOCIETY.
December 8, 1851.
On the Oscillations of Suspension Bridges. By J. H. Rohrs,
Esq., M.A.
In this paper the oscillations of a chain suspended at two points
were discussed, with a view to explain the causes of fracture in sus-
pension-bridges, by vibration arising from the tramping of troops,
gusts of wind, &c., as well as to suggest means for obviating the
mischief under those circumstances. The following were some of
the most remarkable results arrived at : —
1st. That if the tension at the ends of the chain where it is sus-
pended be kept constant by allowing play at those points, the varia-
tion of tension due to vibration at any other point of the chain will
be but small.
2ndly. That if the chain be tied at the points of suspension so
that it can have no motion there, a slight extent of vibration will
produce comparatively a great increase of tension.
3rdly. That periodic forces, such as may be taken, for instance, to
represent the effect of tramping in time of troops moving across the
bridge, are dangerous in the extreme, as if they happen to coincide
in period with any of the possible types of vibration, the extent of
vibration will increase continuously, till it ceases to be represented
approximately by a linear or even an equation of the second order ;
in this case, the chain will be divided by nodal points where there is
no vertical motion.
4thly, That the mere transit, without tramping, of ordinary loads
at an ordinary pace would not cause sensible vibration in a bridge
of wide span ; but that terms not periodic might be introduced by the
variable pressure of wind sweeping in rapid gusts along the platform.
February 16, 1852.
On the Composition and Resolution of Streams of Polarized Light
from different Sources. By Professor Stokes.
In this paper the author investigates the nature of the light result-
ing from the union of several independent streams of polarized light.
No. IX. — Proceedings of the Cambridge Phil. Soc.
116
The refrangibility of the several streams is supposed to be the same,
and the polarization to be of the most general nature, that is, to be
elliptic. The following proposition is established.
When any number of independent polarized streams, of given
refrangibility, are mixed together, the nature of the mixture is com-
pletely determined by the values of four constants, A, B, C, D, de-
fined in the following manner : — Let J be the intensity of one of the
elliptically-polarized streams, a the azimuth of its plane of maximum
polarization, tan /3 the ratio of the axes of the ellipse described by
the sethereal particles ; then
A=S(J); B=S(Jsin2/3); C=2(Jcos2/3 cos2a) ;
D=2(Jcos2/3sin2a).
Two groups of polarized streams, of the same refrangibility, which
are such as to give the same values to each of the four constants
A, B, C, D, are defined to be equivalent ; and the author has shown,
that if two equivalent groups be transmitted through any optical
train, and be afterwards analysed, they will present exactly the same
appearance ; so that equivalent groups may be regarded as optically
identical.
It readily follows from the above theorem, that any group of
polarized streams is equivalent to a stream of common light com-
bined with a stream of elliptically-polarized light from a different
source. If J, J' be the intensities of these streams, a' the azimuth
of the plane of maximum polarization of the latter, tan /3' the ratio
of the axes of the characteristic ellipse,
J=A- V(A2-|-B2-i-a^) ; J'= ^/(A2 + B2.i-C2) ;
sin 2/3'= —— — \ -— ; tan "la'^^.
The author has applied these formulae to a few examples, and has
likewise shown, from the general principles established in the paper,
that the changes which are continually taking place in the epoch and
intensity of the vibrations of polarized light may be of any nature.
In the case of common light, the author contends that there is no
occasion to suppose the transition from a series of vibrations of one
kind to a series of another kind to be abrupt, but that it may be of
any nature.
Professor Miller made a communication on the Artificial Forma-
tion of Crystallized Minerals.
March 1,1852.
Mr. Hopkins, F.R.S. &c., gave a Lecture on the Influence of In-
temal Heat, Stellar Radiation, and Configuration of Land and Sea
in producing Changes of the Earth's Superficial Temperature.
117
March 15, 1852.
Professor Miller made a communication on different improvements
in the Reflective Goniometer ; and a description of a New Reflective
Goniometer.
Professor Stokes concluded a paper on the Composition and Reso-
lution of Streams of Polarized Light from different Sources (see the
abstract under the date Feb. 16, 1852, Phil. Mag. vol. iii. p. 316).
He also made a communication on Haidinger's Brushes.
Also on the Optical Properties of a New Salt of Quinine. The salt
alluded to is that which had recently been discovered by Dr. Herapath
(Phil. Mag. vol. iii. p. 161). The substance of this communication
formed the subject of a notice of the properties of the salt which the
author read at the Meeting of the British Association at Belfast,
which will be found in the Report of the Transactions of the
Sections.
April 26, 1852.
llie Rev. Mr. Kingsley gave an account of the application of Pho-
tography to the Microscope.
The earliest attempts in photography were directed both by Sir
H. Davy and Mr. Fox Talbot to the fixing upon prepared paper the
images of objects by the solar microscope, and the latter gentleman
succeeded completely, as far as his instrument allowed, in obtaining
pictures of minute structures. Shortly after the publication of Mr.
Talbot's process, various attempts were made to apply the oxyhy-
drogen microscope to the same purpose, as that instrument, had
superseded the solar. The result however was, that it vias aban-
doned on account of the great time that was found to be necessary
for impressing an image ; and after a great variety of trials by Prof.
Owen, Dr. Carpenter, Dr. Leeson and others, the use of the instru-
ment for this purpose was given up.
The discovery of the collodion process, so much more sensitive
than that of Mr. Talbot, led the author to think, as soon as he be-
came acquainted with it, that we were in possession of the means of
impressing microscopic objects by means of artificial light without
any great trouble. A friend of his had an oxyhydrogen microscope
of the common form, and on making a trial with it, he found that
by using a very sensitive kind of collodion, he could obtain images
by about a minute's exposure. On examining the instrument, how-
ever, he saw that its form must be completely changed, in fact, that
an entirely new kind of instrument was required to obtain the best
effect. The two points to be regarded as the peculiar principles of
this microscope are, 1st, that none of the radiant light be lost, or as
little as possible ; 2ndly, that the magnifying power be obtained by
such means as would not place the screen for receiving the image
beyond such a distance from the object, that the motions of the
118
instrument could be governed at the same time that the image
was closely inspected. The first of these objects is secured by giving
a very large angular aperture to the system of lenses used for col-
lecting the light, and by using another set of lenses for condensing
it again on the object, and so arranging their focal length in pro-
portion to the focal length of the object-glass, as to cover the plate
to be acted upon, and that space only : the second, by using a sort
of eye-piece for enlarging the image formed by the object-glass.
The lenses divide themselves into four groups, as represented in
the fiu-ures, in which the light is supposed to proceed from the left
hand to the right. The first set for collecting the light is composed
of three large lenses, a meniscus, plano-convex and double convex,
being a combination of three lenses similar in effect to Herschel's
doublet ; the second set for condensing the light on the object is a
similar set of lenses, but of much shorter focal length, and turned
the other way ; between these two sets is a plano-convex or plano-
concave lens placed at its focal length from the convergence of the
rays from the first group, so as to make the rays pass to the con-
densers in a state of parallelism, and so do away with the necessity
of changing the distance between the collectors and condensers
for each adjustment of the latter : the third group forms the object-
glass, which must be so corrected as to have the rays of the spec-
trum between the fixed lines G and H as much as possible brought
to a point, as these rays are those that produce the maximum ac-
tion on the silver salts used in photography ; this will require the
red rays to be left untouched, just in the same way as Fraunhofer
left those of the blue end of the spectrum dispersed in correcting an
object-glass for light. The fourth group is the common eye-piece
left under- corrected. A rather better form for this is a Ramsden's
eye-piece with the first lens partially achromatized, by making it a
compound lens with the radius of curvature of the common surface
aearly double that of the surface that would render it achromatic.
119
This form of eye-piece gives a better correction of the oblique pen-
cils than the common negative.
The time of exposure to obtain an intense negative six inches
diameter, on a collodion plate prepared as below, is about a minute ;
a positive is obtained in a fraction of a second.
The collodion is formed by dissolving gun-cotton in sulphuric
ccther, and adding to it a small portion of iodide of silver dissolved
in iodide of potassium, and also a very small portion of bromide of
iron, or of iodide or bromide of arsenic. The image is developed by
protonitrate of iron, or by a solution of pyrogallic acid in acetic
acid and water, and fixed by a solution of hyposulphite of soda.
By taking out the two first lenses of the collectors, the instrument
is adapted for using sunlight.
Note. — At the time that this communication was made to the
Society, Prof. Stokes had kindly made known to the author the re-
sults of his discoveries with regard to the rendering visible the che-
mical spectrum, but as he had not then made them public, the author
of this communication could not state the use that Prof. Stokes's
discovery enabled him to make of a screen composed of uranium
glass, or of infusion of horse chestnut bark, for finding the focal di-
stance of the chemical image, or of arranging the lenses of the con-
denser so as to produce the maximum of chemical action.
Also, since the communication was made, it has been found that
the instrument described gives light enough to impress an image on
any of the ordinary papers or Daguerreotype plates in periods ran-
ging between one and five minutes, with the oxyhydrogen and lime
light; and with direct sunshine the impression is almost instantane-
ous ; of course sunlight is much better than any artificial light when
it can be procured, both as regards speed and the clearness of the
picture produced.
May 10, 1852.
Professor Miller gave an account of a new method of adjusting
the Knife-edges of a Balance.
Also of a method of determining the height of clouds by night.
May 24, 1852.
Professor Stokes gave a Lecture on the Internal and Epipolic
Dispersion of Light.
November 8, 1852.
Mr. Adams, F.R.S. &c., gave an account of some Trigonometrical
Operations to ascertain the difference of geographical position be-
120
tween the Observatory of St. John's College and the Cambridge
Observatory.
The observations, especially those of eclipses and occultations,
which were made during many years by the late Mr. Catton at the
Observatory of St. John's College, and which have recently been
reduced under the superintendence of the Astronomer Royal, render
it a matter of some importance to determine the exact geographical
position of that Observatory. The simplest and most accurate means
of doing this appeared to be, to connect it trigonometrically with the
Cambridge Observatory. For this purpose, a base was measured
along the ridge of the roof of King's College Chapel, by means of two
deal rods terminated by brass studs, the exact lengths of which were
determined by comparison with a standard belonging to Professor
Miller. The extremities of the base were then connected by a tri-
angle, with a station on the roof of the Observatory at St. John's,
from which, as well as from the two former points, a signal post on
the roof of the Cambridge Observatory could be seen. The angles
at the extremities of the base, combined with the corresponding ones
at the station at St. John's, furnished two determinations of the
distance of the Cambridge Observatory, which served to check one
another. The meridian line of the transit instrument at St. John's
passes through King's College Chapel, so that by observing the point
at which it intersected the base, the azimuths of the sides of the
triangles could be immediately found.
The result thus obtained is, that the transit instrument of the
Cambridge Observatory is 2313 feet to the north, and 4770 feet to
the west of that at St. John's College. Hence it follows that the
difference of latitude is 22"-8, and the difference of longitude 5"'10 ;
and the latitude of the Cambridge Observatory being 52° 12' 51""8,
and its longitude 23""54 east of Greenwich, we have finally for the
geographical coordinates of the Observatory of St. John's College,
Latitude.. 52° 12' 29"-0
Longitude O'' 0' 28"- 64 E. of Greenwich.
These operations, of course, furnish incidentally, a very exact
determination of the orientation of King's College Chapel. The line
of the ridge of the roof points 6° 20'-3 to the north of east.
November 22, 1852.
Professor Challis made a communication on the recent return of
Biela's Double Comet.
December 6, 1852.
Professor Stokes gave an account of M. de S^narmont's Researches
relating to the Doubly-refracting Properties of Isomorphous Sub-
stances.
121
February 7, 1853.
An addition was read to a paper by Professor De Morgan on
the Symbols of Logic, the Theory of Syllogism, &c.
A paper was read by Mr. Denison on some Recent Improvements
in Clock Escapements.
The object of this paper was to explain the construction of a new
remontoire or gravity escapement invented by the author, which has
now been in action for some time on the pendulum of the great
clock for the houses of parliament, and is in course of application to
others, both turret clocks and astronomical.
But by way of introduction to this, which may be called the three-
legged gravity escapement (from the form of the scape- wheel), Mr.
Denison gave a description of another, which would similarly be
called the three-legged dead escapement, and had been previously in-
vented by him for the purpose of giving the impulse to the pendu-
lum with far less friction than usual. He found that it required only
^th of the force which a common dead escapement had required to
make the pendulum swing the same arc. And therefore, as com-
pared with a gravity escapement in which there is no sensible fric-
tion on the pendulum, there must be still more than fths of the
force in a common dead escapement wasted, in first producing fric-
tion on the pendulum, and then overcoming it by an iucreased im-
pulse. The time of the pendulum would be much more disturbed
than it is by the inevitable variations of this large amount of friction,
as well as that of the clock train, but for a fortunate tendency of the
different errors, which are caused by these variations of force and
friction, to correct each other.
But the amount of this self-correction is uncertain, and some-
times one set of errors preponderates and sometimes the other ; and
so a dead escapement clock sometimes gains and sometimes loses
simultaneously with either an increase or a decrease of the arc of
vibration. And, consequently, none of the contrivances for iso-
chronizing a pendulum for different arcs can secure isochronism of
the clock ; and no further material improvement in clocks can be
expected, but from the solution of what has long been known as the
great problem of clock-making, viz. the invention of a simple escape-
ment which will give a constant impulse to the pendulum without
any sensible friction.
Mr. Denison showed that his new gravity escapement satisfies all
the requisite conditions, mechanical, mathematical, and oeconomical.
Its principal features are, that the scape- wheel has only three pins,
not far from the centre, which lift the pallets or gravity-arms, and
three long teeth which are locked by stops on the arms. The velo-
city of the scape-wheel, which usually produces tripping, if the force
of the train is increased beyond what is just enough to lift the arms,
is moderated by a fan-fly set on the axis of the scape- wheel. The
arms are necessarily longer in this than in any other gravity escape-
ment, and this also gives a greater depth of locking within a given
angle, and therefore a still further security against tripping. And if
an ai"m is by accident lifted a little too high, the tooth does not
122
escape, and the arm falls down again to its proper height until the
pendulum carries it oiF, the pressure of the long teeth on the stops
not being enough to hold it up. For these reasons also there is no
difficulty in satisfying the mathematical condition investigated by
Mr. Denison in a paper read before the Society in 1848, viz. that y
(the angle at which the pendulum leaves one arm and takes up the
other) should = " , or at any rate not be less than - (a being the
V 2 3
extreme arc of vibration). The escapement requires no oil in the
parts aflfecting the pendulum ; and it contains no delicate work, and
is very easy to make ; and as a highly finished train will be no
longer necessary, astronomical clocks may be made on this plan much
cheaper, as well as better, than heretofore.
In turret clocks an escapement of this kind supersedes the
necessity for a remontoire in the train to equalize the force on the
scape- wheel, and also of long and heavy pendulums, which are expen-
sive when compensated, and are sometimes difficult to fix. It will
also allow cast-iron wheels to be used throughout the clock (which
Mr. Dent has now used for several years in connexion with Mr.
Denison's spring remontoire for the train), as the friction of the train
can no longer affect the pendulum.
February 21, 1853.
Professor Challis gave a Lecture on Halos, Parhelia, and Para-
selense.
March 7, 1853.
Professor Stokes gave an account of some further researches
relating to the Change of Refrangibility of Light.
April 11, 1853.
The Rev. Mr. Pritchard, F.R.S., gave an account of the Processes
requisite to render Quicksilver tremorless for Astronomical Obser-
vation.
The great improvements recently introduced, and especially by
the present Astronomer Royal, in the construction and methods of
using astronomical instruments, require a far more extended use of
reflexion from mercury than heretofore. Unfortunately, however,
both the convenience and the accuracy of these methods have been
greatly limited and impaired by the tremors to which mercury is
liable. Many attempts have been made both in France and in Ger-
many to remove or obviate these tremors, but hitherto by no means
with perfect success. The Rev. C. Pritchard, of Clapham near
London, has proposed a method which appears fully adequate to the
requirements of astronomy. It consists in the adoption of a silver-
plated or amalgamated copper vessel of a peculiar form, admitting
the use of a very thin stratum of mercury without the necessity of
an inconvenient amount of shallowness in the vessel itself. Mercury,
123
however, placed in an amalgamated vessel after a short time becomes
covered with a singular film of amalgam, which impairs the reflect-
ing power of the surface, and if at all agitated, soon entirely destroys
it. And this is the case even when the vessel is made of amalga-
mated platina. The most important, and by far the most difficult
part of Mr. Pritchard's experiments, consisted in the invention of a
method by which these films can be easily and practically removed.
The details, many of which are curious and interesting, would here
occupy too much space, but they are fully explained in a memoir
recently read to the Royal Astronomical Society of London ; and it
may be added, that the process has been adopted at the Royal Obser-
vatory at Greenwich, and is now in progress of trial at the Observa-
tories of Paris and Cambridge.
April 25, 1853.
Professor Challis gave a lecture on the Adjustments of a Transit
Instrument.
A paper was read by Professor De Morgan on the Principle of
Mean Values, and an addition to a paper on the Symbols of Logic,
&c. in vol. ix. part 1 of the Society's Transactions.
Though the heading of this paper describes one of its main results,
yet it might with equal proJTriety have been styled a discussion of
some points of algebra, with reference to the distinction of form and
matter. This distinction, it is contended, is more extensively applied
in algebra than in logic, though more recognized in logic than in
algebra. Looking at the disputed points which exist in the higher
parts of mathematics, and feeling satisfied that they will never be
settled until the separation of form and matter is both visible and
complete, the author makes a first attempt towards the examination
of the question how far this yet remains to be done. A number of
comparisons are made between algebraical and logical process, in
the course of the inquiry, illustrative of the opinion entertained by
Mr. De Morgan, that logic, as treated, requires the interposition of
the algebraist, and cannot, except by aid of algebraical habits, be
rendered a complete exposition of the forms of thought. In digress-
ive notes, he combats the opinion that a generalization of the quan-
tity is, as £isserted, a new material introduction. He argues against the
too mathematical tendency of some of the logicians who have endea-
voured to extend the ancient system, especially the attempt of some to
make the logical import of the proposition noMen^ but a comparison
of more and less, and an equation or non- equation of quantities. He
points out that the proposition has been formalised in nothing but
its terms, subject and predicate ; and gives an instance of the method
in which a failure of general maxims is answered by the sole asser-
tion that the mode of expression which brings about the failure is
useless. He refers to what were called sophisms, contrasting the
neglect of them by the logicians with the use which the algebraists
have made of their corresponding difficulties, as in the case of nega-
tive and imaginary quantities, the fraction §, &c. He argues against
the assertion of more than one eminent writer on logic, that the
124
identity of two terms, X and Y, expressed as in " all X is all Y," is
not a complex proposition — is not the union of Every Y is X with
Every X is Y. In an appendix to a former paper on the symbols of
logic, he refers to a complaint of misrepresentation made by Sir W.
Hamilton of Edinburgh, to whom certain technical phrases had been
attributed. Mr. De Morgan makes the requisite correction, affirms
that he had good reason for attributing such phraseology, and points
out what that reason was : he then proceeds to answer two new
charges of plagiarism against himself, from the same quarter ; giving
as his reason for addressing such answer to the Society, that Sir W.
Hamilton makes the appearance of the asserted plagiarisms in the
Transactions his principal ground of notice.
Finally, as to the logical part of the communication, Mr. De
Morgan, reverting to his complex syllogism, in which each premise
and the conclusion contain two ordinary propositions, generalizes the
premises into the numerical form, and, giving terms and quantities
algebraical designations, points out the mode of producing all pos-
sible inference. The immediate occasion of this introduction is as
follows : — Sir W, Hamilton, in a recent publication, one tract of
which is directed against Mr. De Morgan's last paper on syllogism,
affirms that a proposition, as to its logical force, is merely an equa-
tion or non-equation of quantities, from which the declaration of
coalescence or non-coalescence of terms into one notion is a conse-
quent. Mr. De Morgan maintains the converse ; namely, that the
proposition is a declaration of coalescence or non-coalescence, of
which the equation or non-equation of quantities is an essential. In
treating the complex syllogism, under definitely numerical quan-
tities, he has to search for the properties of the equation of coales-
cence, as distinguished from the equation of quantity ; and, having
made the former the means of arriving at inference, he invites those
who can to try if the same result can be produced by means of the
latter alone.
To pass to the algebraical part of the paper. It is first contended
that the states infinity and zero, whether represented by distinctive
symbols attached to and oo , or by negative and positive powers of
dx, must be formally distinguished, as being each, not a value, but
a status, containing an infinite number of corrational values, just as
happens in finite quantity. In order to lay down the formal laws of
connexion of these different states, it is necessary to examine the
formal use of the symbol = . After pointing out instances in which
the laws of algebra are by many declared invalid, as by those alge-
braists who admit and interpret '2x=^x, but cannot give permission
to divide both sides by x, the following laws are suggested. The
symbol = is to be read with an index, as in =„, which has reference
to the order oo„ or 0_„, or as in =_„, which has reference to oo_„
or to 0„. The equation A=raB is normally satisfied when A and B
are of the order n, and A— B of a lower order. It is supernormally
satisfied if A and B be both of any (the same or diff'erent) higher
order than the wth, and suhnormally if both be of any lower order.
Among the most conspicuous rules which follow.are that AC=;„+,jBD
is normally satisfied, if A=;„B and C=«D are so ; and that when an
125
equation is multiplied or divided by a quantity of the order n, the
index of equality must be increased or diminished by n. Various
cases are given in which such results as now present anomalies
are reduced under formal law, and others which would be absolutely
rejected are shown to be capable of consistent interpretation.
The formal law of connexion of the different states, of which fini-
tude (with the index 0) is only one, is that the order 0^ stands to
finite quantity in all respects as finite quantity to oo^ . Hence, so far
as 1 and 1 + are simultaneous as well as equal, so fur oo and oo + a
are simultaneous as well as equal. And if^(l)=o0(l+O) be a uni-
versal law, so must be 0(oo) = o^(oo -\-a). Further, oo — oo must
be, formally speaking, wholly indeterminate, even when it is a case
of X — X.
In relation to such indeterminate forms as oo — oo , §, &c., Mr. De
Morgan contends that their formal and a priori character is that of
indeterminateness ; and that the choice between determinate and in-
determinate character, which so often occurs, is dictated by the matter
of the problem, the determinate value being dictated by the laws of
algebra. The index of equality, for instance, may be the means of
decision : an example is given in which one equation belongs to two
different problems, but with different indices of equality ; in one §
is determinate, in the other wholly indeterminate.
In assigning oo or as values, it is often necessary to assign rela-
tions of order. When a quantity passes from positive to negative,
or the converse, through or oo , it passes through every order of
or 00 ; and this even when the passage is from one phase of or co
to another, of different signs. Thus, the orders being powers, x
cannot pass from — a.O"* to +0.0"*, without passing through
even 0=c .
Mr. De Morgan insists upon one of two things : either, the aban-
donment of the separate use of and oo, except only in the retention
of the former symbol to represent A — A ; or, the introduction of dif-
ferent orders, and the free use of the comparisons of those orders.
For himself, he prefers and adopts the latter alternative.
The principle of limits is considered as a formal law of algebra,
but not to the exclusion of every other result. If a constant, for
instance, have the value A up to xz=.a exclusive, it has A for one
value when x=za. If the constant be transitive, that is, if it be
always =B after x-=a, then x=a gives both A and B for the con-
stant, and, as ^fact hitherto observed, its value from calculation is
|^(A+B). This observed fact Mr. De Morgan believes he connects
with the principle of limits, making it a necessary consequence of
the universal truth of that principle ; and hence he holds that it may
be stated as a theorem, under the name of the principle of mean
values. Various uses of this principle are given. Further, in assu-
ming the free use of the orders of and oo , it is shown that it is cor-
rect to say that the constant passes from A to B while h, x being
a + h, passes through the phases of 0. So that, for instance, at an
epoch of transitiveness the value of
then dA^=Ay(y' — x)*^*^' ^^^ X— — A^. : A^ are identical equations.
Every relation which satisfies Ay=oo is a solution, and a singular
solution ; except possibly, relations of the form x= const., which
must always be examined apart. Also, A^=oo is identical with
i^g=0. There can exist no solutions whatsoever except those which
are contained in A=const., Aj,=oo, and (possibly) x=const.
Again, x«~(^°S^a)** ^^ ^^^^ equation the author has found
neither notice nor use : supposing it to have ever been given, he
holds it most remarkable that it has not become common as the mode
of connecting the two well-known and widely used tests of singular
solution. It easUy shows that Xi^=oo contains all extraneous solu-
tions, and all intraneous solutions which (as often happens) can be
also obtained by making a a function of x. It also easily gives a
conclusion arrived at by the author in his last paper, namely, that
when j^ =00 is satisfied and not y'=X' ^^ follows that x, + x«X ^*
infinite.
7. The author gives his own version of the demonstration of a
theorem of M. Cauchy, for distinguishing extraneous and intraneous
solutions. If y =P, P being a given function of x, satisfy y'=^x(^' y)>
that is, if P' and x(^i P) be identical, then y=P is an extraneous or
138
intraneous solution of y'=x(a?, y), according as
•P+^ dy
i:
x(*»'y)-x(^>P)
{x being constant) is finite or infinite for small values of /?. This
theorem has attracted little notice in this country : the author
believes it to be fully demonstrated, and considers it one of the most
remarkable accessions of this century to the theory of diflferential
equations.
8. It is observed that the validity of the extraneous solution may
depend upon the interpretation of the sign of equality by which A =B
is held satisfied when both sides are 0, or both infinite, even though
A:B=lisnot satisfied. Thus y'^-lVy or y=^{x-{-aY, has the
extraneous solution y=0, which, however, is not a solution if by
y'=2 Vy we understand in all cases y' : 'v/y=2.
9. The common mode of obtaining the singular solution of a bior-
dinal (by combining ^(.r, y, a, b)=0, da, b

a'Pbx\y — ^Ay. y' = Yb' -^Xb' ;
the biordinal factors, y" — \{x,y,y'), b"—a(a,b,b'), disappearing
, 140
from b' and y'. Hence b' depends on x, y, y'. Similarly, b" depends on
X, y, y', y", &c., and similarly for y',y",&.c. If inf(x,y, y',y",&c.)=0
we substitute for x, y, y', &c. in terms of a, b, b', &c., the two equa-
tions belong to polar reciprocals. If either can be integrated, the
integration of the other depends on elimination : thus if the equa-
tion in a, b, &c. can be integrated, the solution of the equation in
X, y is obtained by eliminating a and b between the integral obtained
and x=^, y=Y.
13. There are two reciprocal biordinal equations belonging to the
modular equation ^(a;, y, a, b)=0 ; y"=x "when a and b are constant,
b"=:a, when x and y are constant>^ The two have the same condition
of singular solution ; for Ay ^j=X4'^^. Let this be a(x, y, a, 6)=0,
when cleared of y' or b'. The following table exhibits the relations
of the double system : —
I (f){x,y,a,b)=0 ,
a{x,y,a,b)=0 \ 0^+0,;y'=O (f)a-^ (x, a, b, c) z='i^(x, a, b, c),
give one, and only one, primordinal of the form /(a;, y, z,y', z')=0.
Assume any surface w{x,y,z)=:0 ; by this, and compensative rela-
tions between a, b, c, another pair of primitives may be found. But
the primitives obtained from w=0 do not shade this surface, except
in cases determined by two relations between the constants. Again,
making a, b, c compensative, without any assumed surface, we find
141
one equation of the form (a, b, c, a', 6')=0, any primitives of which
lead to other primitive forms fory^O. Each of the second primi-
tives has contact of the first order with one family of curves from
among the original primitives ; and all ordinary primitives are found,
in an infinite number of ways, among the connecting curves of others.
There is a singular solution, a curve of contact to all primitives, when
4>^=0, '^^'=0, &c. can all be satisfied at once.
Since y=<&, r=^, give a primordinal equation independent of
constants, the polar reciprocal properties of curves in space are of a
restricted form. Every surface dictates another surface, and a mode
of shading both, so that each line of shading on either surface is the
polar reciprocal of a line on the other.
16. The conversion of constants into compensative variables may
give restricted solutions, as in the ordinary case of two variables, and
every other in which the constants are converted into separately
self- compensating variables. When these variables are made collect-
ively compensating, and the equations permit elimination of the
original variables, ordinary differential equations may be produced,
the integration of which may, after substitution, give primitives of
the same form as those from which they came. But when the ori-
ginal variables cannot be eliminated, arbitrary relations may be
required, in number enough to eliminate the differentials of the new
variables : in this case arbitrary functions enter the jirimitives finally
deduced. Of this last case one instance is Lagrange's transition
from a primitive of a primordinal partial equation having two con-
stants to the complete primitive of that equation.
17. A biordinal partial equation may be produced from
lJ{x, y, 2, a, b, c, e, A)=0
by eliminating the five constants between U=0 and the five results
of primordinal and biordinal differentiation. But it is not true that
every form of U=0 leads to one biordinal equation only : many
forms lead to an infinite number. Two attempts to procure other
primitives by making a, b, &c. compensative variables, end in two
different forms of result. First, when all the resulting equations
are required to be integrable, by introduction of a proper factor, the
success of the process requires the integral of two partial equations,
,one primordinal and one biordinal, between four variables. Secondly,
when no such condition is required, the result is another form in-
volving five constants.
18. A primordinal partial equation belongs to a family of surfaces
of which one is determined by any given curve through which it is
to be drawn. A biordinal equation belongs to an infinite number of
families ; and a distinct conception of the conditions which select
an individual surface is best formed by an extension of the following
kind. A curve on a surface is analogous to a point on a curve : two
curves being drawn on a surface, the analogue of the chord joining
two points on a curve is the developable surface (or surfaces) drawn
through the two curves. The developable surface which touches
the given surface in a curve (and not the tangent plane) is the ana-
142
logue of the line which touches a curve in a point. A biordinal
equation being given, one surface satisfying it is selected by a curve
through which that surface is to pass, and a developable surface
passing through that curve which the surface is to touch.
19, 20. The restrictions under which two arbitrary forms must
enter, in order that a biordinal partial equation may exist indepen-
dent of these functions, are wholly unknown. The case which is
fully analogous to a biordinal of two variables, is of the most limited
character. Ampere has noticed this : Mr. De Morgan was led to
it by an examination of the polar properties of ^(a;, y, z, a, b, c)=0.
This equation leads to a=A, b=B, c=C, where A, B, C are func-
tions of X, y, z, p, q. The primordinal/(A, B, C)=0 is satisfied by
^=0, subject to (x,y, z, a, b, c) =
by means of x, y, z and of a, b, c, are connected by relations analo-
gous to those already seen in the case of two variables. In fact,
there is perfect coincidence and coextension between the properties
of the general equation y"= j^(a?, y, y') and a particular species of the
equation Q+Rr+S5+T;4-U(s2_r^)=0. It is proposed to call
this species the polar biordinal.
21. The general method of transforming partial equations, given
in the last paper, is the investigation of the class of surfaces con-
tained under a given equation by reference of them to their polar
reciprocals, any convenient modular equation ^{x, y, z, a, b, c) being
made the means of transformation.
22. The following notation is proposed for eliminants. ITie com-
ponents being A^, A^, &c., B^, &c., the eliminants are (Aj^j,(AB^^^,
(ABCpqr), &c. ; the components being A, A', &c. B, &c., the elimi-
nants are (A°), (AB° '), (ABC° ' "), &c. Thus
(Ap)=Ap
(ABp5)=Ap(Bj)-Bp(A,;
(ABCpgr)=Ap iBCgr)-\-Bp (CAjr) + Cp (AB^,)
(ABCDp,,,)=Ap(BCD5,,) -BpCCDA,^) +Cp(DABgrs) -T>p(ABCgrs),
and so on. Some slight investigation of properties is made, to ex-
hibit the notation.
143
The following rule is suggested to determine, in any complicated
case, whether the number of contiguous interchanges by which one
arrangement of letters is converted into another shsdl be odd or even.
This is an important matter in the theory of eliminants, though very
complicated instances may seldom occur in practice. Write down
one arrangement under the other, and, beginning at one letter in one
line, mark the companion letter in the other line, pass on to that
companion in the first line, mark its companion, and so on, until we
arrive at a letter already marked. Call this sequence a chain, each
mark being one link. Having formed one chain, begin at a letter
not yet used, and form another; and so on until every letter has
been used. Then, according as the number of chains with even
links is odd or even, the number of interchanges of contiguous letters
required is odd or even. For example, the two arrangements being
ABCDEFGHIJKLMNOPQ
HMOGQBKLJPFC I NADE
121 23221222124 12 3.
Under A is H, under H is L, under L is C, under C is O, under
O is A, already taken : the first chain has five links, the second is
found to have nine, the third two, the fourth one. The number
having even links is one, an odd number ; hence an odd number of
contiguous interchanges converts the first arrangement into the
second.
23. The following is the method of ascertaining whether the bior-
dinal equation
Q + Rr + S5+T/+U(5«-r/)=0 . . . . (1)
possesses a primordinal of the form f(x,y, z, p, $')=0, containing
an arbitrary function. Considering x, y, z, p, q as five independent
variables, integrate, by common methods, the equations
dv\ . Twydv k ^dv „
\dy dz/
l + k dq
dv\ . T, dv 1 Q,dv
dq l+k dp
k being one of the roots of ArS^ = ( 1 + A)2(RT + QU) . If a common
solution i;=A can be found, then A=const. is a primordinal of (1).
If two common solutions, A and B, can be found, then B=orA is
a primordinal, w being arbitrary. But though in this case A=const.
and B= const, are solutions, they ctmnot coexist, unless the values of
k be equal, or unless S*=4(RT + QU). This last equation is one
condition of polarity ; and if, when satisfied, we find three (and there
cannot be more) common solutions. A, B, C, inexpressible in terms
of each other, then /(A, B, C) = is the most general primordinal,
any two forms of it may coexist, or even any three, which amount
to A = const., B= const., C = const. Elimination of ja and q between
these last equations gives ^(x, y, z, a, b, c)=0, the modular equa-
144
tion. And the general solution of (1) is found by assuming b and
c in terms of a, and then making a a self- compensating variable,
24. The paper is concluded by some remarks on notation.
In an appendix to the preceding paper, read to the Society on the
1st of May, 1854, Mr. De Morgan points out an error committed
by M. Cauchy in a very remarkable theorem, of which his enuncia-
tion is as follows.
Let cpx be a function which can be developed in integer powers
of a;. Let r(cos 6 + sin0 . ^/ — 1), r being positive, be any one of
the roots of 0x= 00 or of ^'a:':=oo . Then the development of ^a? is
convergent from a:=0 up to a; = the least value of r.
M. Cauchy stipulates that the function shall be continuous ; but
he defines a function to be continuous so long as it remains finite,
and receives only infinitely small increments from infinitely small
accessions to the variable. It is then obviously impossible that the
above theorem should be universally true. Were it so, it would
follow that the development of (l+^r)^ is convergent for all finite
values of x, whereas it is well known that this development becomes
divergent when x is greater than unity. The error of M. Cauchy's
demonstration (which contains a valuable method for establishing a
large class of definite integrals) is the assumption that if an infinite
number of convergent series of the form a + bx-\-cx'^-\- . . . , ^\\ with
one value of x but different values oi a, b, c, . . . , be added together,
the sum divided by the number of series is also a convergent series.
This assumption is not universally true.
Mr. De Morgan takes a totally different line of deroonstratron,
and establishes the following theorems.
If r(cos0 + sin0. a/ — 1), r being positive, represent a root of
any one of the equations ^t=oo , 0'd?=co , (j)"x-=oo , . . . . then the
development of cpx in powers of x is always convergent from x=0
up to x=: the least value of r, and divergent for all greater values oi x.
If the development have all its coefficients positive, or if all beyond
an assignable coefficient be positive, the least value of r is obtained
from a real and positive root.
If the signs of the development be, or finally become, recurring
cycles, with / in each cycle, the least value of r is obtained from a
root in which cos + sin . V—l is one of the lih roots of unity.
If no such cycle be finally established, cos + sin0 . V—l may have
a value of 6 which is incommensurable with the right angle,
M. Cauchy has established from his own theorem (the want of
sufficient statement of conditions not affecting this particular case)
the necessity of the observed fact, that the developments produced
by Lagrange's theorem for the development of implied functions
always give, when convergent, the least of the real values which are
implied.
145
May 1, 1854.
A paper was read by Professor De Morgan on the Convergency of
Maclaurin's Series, being an Appendix to a paper on some Points in
the theory of differentisJ equations. See the abstract of the former
paper, Phil. Mag. vol. vii. p. 450.
Mr. Kingsley made an oral communication on the Chemical
Nature of Photographic Processes,
May 15, 1854.
A paper was read by Mr. Warburton on Self-repeating Series.
In computing Bernoulli numbers by the formula of Laplace*, the
author of this paper was led to notice, that in the fraction whose
development is a series of the form i2)»+i_22n+i^_j_g2«+i ^ fi_^f,^
the numerator of that fraction is a recurrent function of t. This led
him to investigate the question, what are the conditions which the
denominator of the generating fraction, and the terms of the series
generated, must satisfy, in order that the numerator of such a frac-
tion may be a recurrent function of t. The paper contains the result
of that investigation.
The author calls those series " self -repeating ," which, when ex-
tended without limit in opposite directions, admit of separation into
two similar arms, each arm beginning with a finite term of the same
magnitude. Between this pair of finite terms, either no zero-term,
or one or more zero-terms, may intervene. One arm repeats, and
contains arranged in reverse order, the terms of the other arm, either
all, or none, of the terms having their signs changed. The different
positive integer powers of the natural numbers, of the odd numbers,
and of the figurate numbers of the several orders, present familiar
examples of self- repeating recurring series.
The author demonstrates the following three theorems respecting
self-repeating recurring series : —
I. If the series arising from the development of a proper fttiction
is the right arm of a self- repeating recurring series, and if the deno-
minator of such a fraction is a recurrent function of t, then the nu-
merator also is a recurrent function of t.
II. Other things remaining the same, if the numerator of the
fraction is a recurrent function of t, then the denominator also is a
recurrent function of t.
III. If the numerator and the denominator of a proper fraction are
each a recurrent function of t, then the series, arising from the deve-
lopment of the fraction according to the positive integer powers of t,
will be the right arm of a self-repeating recurring series.
By way of example, the author applies his first theorem to the
summation of the infinite series 1''— 2^-|-3^ — &c., and compares his
process with the corresponding processes of Laplace and of Sir John
Herschel. The sum in question is given by Sir John Herschel (see
Jameson's Journal, January 1820) in terms of the differences of the
powers of 0, extending from A'O^ to A^O^. In the author's process,
* See Memoirs of the Academy of Sciences, 1777-
146
the requisite differences extend from A'O^ only to A^O", and the nu-
merical coefficients of these are of diminished magnitude, and of
very easy determination.
The author makes other applications of his theorems; but on
these we forbear to enter-
A paper was read by Professor Challis on the Determination of the
Longitude of the Cambridge Observatory by Galvanic Signals.
The experiment of which this paper contains the details, was
made at the suggestion of the Astronomer Royal, and conducted
according to a scheme arranged by him for giving and receiving
the signals. A galvanic connexion having been established between
the Greenwich Observatory and the Cambridge Telegraph Office,
by means of the Londqn central station of the Electric Telegraph
Company, signals were sent on the nights of May 17 and 18,
1853, between 11** and 12^ mean time. The signals were made
by causing two needles, one at Greenwich, the other at Cam-
bridge, to start by completing the galvanic circuit at either place of
observation. The times of starting were noted at both places, and
reduced to the sidereal times of the respective observatories, to serve
by comparison for determinations of the difference of their longi-
tudes. On each night the signals were made alternately for a quarter
of an hour at one station, in batches containing an arbitrary number
of signals not exceeding nine, and then for a quarter of an hour at
the other station in a similar manner. On the first night the total
number of signals was 151, and on the second night 139. The two
observers, Mr. Dunkin of the Greenwich Observatory, and Mr. Todd
of the Cambridge Observatory, changed places in the interval between
the two nights' observations ; Mr. Todd observing at Greenwich, and
Mr. Dunkin at Cambridge, tn the second night. Also it was arranged
that the two observers should observe identical stars on the two
nights, as well as the stars ordinarily used for clock errors, and that
the same apparent right ascensions of the stars should be employed
for reducing the signal-times at both observatories. The Cambridge
Observatory time was conveyed with the greatest care to the Tele-
graph Office at the Cambridge Railway Station by the transfer of
three chronometers. By a first calculation, the longitude of the
Cambridge Observatory was found to be 23*'03 east of Greenwich.
Professor Challis subsequently made another calculation, taking
into account the effect on the times of meridian transits of stars
produced by the forms of the transit-pivots, according to a method
which he has described in the Memoirs of the Royal Astronomical
Society (vol. xix. p. 103). The errors arising from the deviation of
the pivots from the cylindrical form being eliminated, the longitude
is found to be 22^-70 east of Greenwich, which is less by 0«-84 than
the value hitherto adopted.
May 29, 1854.
A paper was read by Professor Fisher, entitled '" Additional Ob-
servations on the Development of the Vertebral System."
PROCEEDINGS
CAMBRIDGE PHILOSOPHICAL SOCIETY.
November 13, 1854.
A paper, by R. L. Ellis, Esq., was read, entitled " Remarks on the
Fundamental Principle of the Theory of Probabilities."
Also, " On the Purbeck Strata of Dorsetshire." By the Rev. O.
Fisher.
The object of this paper was to describe the beds from which a
series of insect remains and other fossils had been collected by the
author, and presented to the Woodwardian Museum.
The connexion of the Purbeck beds with the Oolitic rather than
with the Wealden series was maintained, while both were shown to
be unconformable in this district to the cretaceous system. Reasons
were given for thinking that the materials, of which both the Wealden
and Purbeck were composed, had travelled from west to east ; and
the beds of the New Red Sandstone, as they occur in Devonshire,
were pointed out as affording a mass of strata which would furnish
a detritus of the character of a large portion of the Hastings sands
of Hampshire and Dorsetshire.
In describing the Purbeck beds.the author followed the system of the
late Professor E. Forbes, dividing them into upper, middle, and lower;
and entered into some detail of the alternations of salt and freshwater
conditions that prevailed during their deposition. The aspects under
which the same beds appear at diflferent points of the district under
examination were particularized, and it was attempted to be shown
that these were in conformity with the theory of a current setting
from the west towards the east. The mode of occurrence of the
remains of insects in the middle and lower Purbecks was somewhat
minutely described, and it was suggested that some interesting
chronological speculations might be grounded upon it.
The paper concluded with an attempt to explain the singular frac-
tured condition of about thirty feet of the lower Purbeck strata
throughout the eastern part of the county. It was supposed that
this might have been caused by the deposition of sediment upon the
remains of the Portland forest before the mass of the trees had been
removed by decomposition ; the sediment, after it had become con-
solidated, settling unequally as the carbonaceous matter was gra-
dually removed.
No. XI. — Proceedings of the Cambridge Phil. Soc.
143
November 27, 1854.
Prof. Willis gave an account of a new form of Atwood's Machine.
December 11, 1854.
A communication was made by Dr. Paget on a case of involuntary
tendency to fall forwards.
February 19, 1855.
Mr. Hopkins gave a lecture on certain changes of Terrestrial
Temperature, and the causes to which they may be attributed.
March 5, 1855.
Dr. Clark gave an account of some recent discoveries respecting
the origin, migrations, and metamorphoses of Entozoa, and their
bearing on the notion of spontaneous generation.
April 23, 1855.
A paper was read by the Master of Trinity, on Plato's Survey of
the Sciences, contained in the seventh book of the Republic.
Plato, like Francis Bacon, took a review of the sciences of his time ;
and like him, complained how little attention was given to the phi-
losophy which they involved. The sciences which Plato enumerates
are arithmetic and plane geometry, treated as collections of abstract
and permanent truths; solid geometry, which he "notes as deficient"
in his time, although, in fact, he and his school were in possession
of the doctrine of the " five regular solids ; " astronomy, in which he
demands a science which should be elevated above the mere know-
ledge of phsenomena. The visible appearances of the heavens only
suggest the problems with which true astronomy deals ; as beautiful
geometrical diagrams do not prove, but only suggest geometrical
propositions. Finally, Plato notices the subject of harmonics, in
which he requires a science which shall deal with truths more exact
than the ear can establish, as in astronomy he requires truths more
exact than the eye can assure us of. It was remarked also, that
such requirements had led to the progress of science in general, and
to such inquiries and discoveries as those of Kepler in particular.
May 7, 1855.
The Master of Trinity read a paper on Plato's notion of Dialectic.
At the end of the survey of the sciences contained in the seventh
book of the Republic, which was the subject of a paper at the
last meet ng, Pkxto speaks of Dialectic as a still higher element of
a philosophical education, fitted to lead men to the knowledge of
real existences and of the Supreme Good. Here he describes Dia-
lectic by its objects and purpose. In other places Dialectic is sj)oken
of as a method or process of analysis ; as in the ' Phaedrus,' where
Socrates describes a good dialectician as one who can divide a subject
according to its natural members, and not miss the joint, like a bad
cancer. Another Dialogue, in which there are examples given of
dividing a subject, is the Sophistes, where many examples of dichoto-
mous or bifurcate division are given. But this ajjpears from the
Dialogue to have been a practice of the Eleatic rather than of the
Platonic school. Aristotle proposed a division of subjects according
to his ten Categories, which he and others since have extensively used.
Xenophon says that Socrates derived Dialectic from a term implying
to divide a subject into parts, which Mr. Grote thinks unsatisfactory
as an etymology, but which has indicated a practical connexion in
the Socratic school. The result seems to be, that Plato did not
establish any method of analysis of a subject as his Dialectic ; but he
conceived that the analytical habits formed by the comprehensive
study of the exact sciences, and sharpened by the practice of dia-
logue, would lead his students to the knowledge of first principles.
Also, Mr. Maxwell gave an account of some experiments on the
mixture of colours.
May 21, 1855.
A paper was read by Mr. Hopkins on the External Temperature
of the Earth and the other Planets of the Solar System.
We have not sufficient data to determine the superficial tempera-
ture of any planet besides our own. We know, however, that it
must mainly depend on the temperature of the planetary space, and
on the heat which the nearer planets at least receive directly from
the sun, but modified, and possibly in a far greater degree than
has been generally supposed, by the particular circumstances by
which each planet may be characterized. The modifying circum-
stances more particularly referred to in this paper, are the existence
of atmospheres surrounding the planets, the positions of their axes
of rotation, and the conductivity and specific heat of the substances
forming the outer crust of each planetary body of our system. No
astronomer, judging from the appearances which Mars and Jupiter
present to us, would entertain any serious doubt as to the existence
of atmospheres surrounding those planets, and the probability would
seem to be almost equally strong of Saturn being likewise enveloped
150
in a similar manner. The obliquity of the axis of rotation is known
•with considerable accuracy in the cases of Mars and Jupiter ; and
also in that of Saturn, if it coincide with the axis of rotation of his
ring. Venus presents great difficulties to the observer, but it ap-
pears now to be pretty satisfactorily determined, that the period of
rotation about her own axis is nearly the same as that of the Earth,
and that the obliquity of her axis is large, amounting to as much as
about 75°. This must produce an extraordinary difference between
the changes of annual temperature in that planet and those which we
experience. The author has endeavoured, in this paper, to estimate
numerically the effect of this anomalous obliquity. Practical astro-
nomers have entertained the opinion that Venus likewise has an
atmosphere. Of Mercury we know too little by direct observation
to form any opinion on those points founded on observed facts, and
the same remark will apply to the remoter planets beyond Saturn ;
but most astronomers probably feel much the same conviction that
Mercury, Uranus, and Neptune have atmospheres of greater or less
extent, as that they revolve round their own axes with greater or
less angular velocity.
It is not the author's object, however, to adjust the balance of pro-
babilities for particular hypotheses in favour of planetary atmospheres
or against them ; but assuming their existence, to estimate their effects
on the planetary temperatures. And in like manner he points out rhe
influence which must be exercised by a greater or less conductivity, and
specific heat in the superficial matter of a planet, without professing to
discuss the probability of such properties being materially different in
the different planets. The Earth's atmosphere is known to be almost
completely diathermanous for heat radiating directly from the Sun ;
and it is assumed to be equally so for the heat which proceeds directly
from the fixed stars, and to which the general temperature of space
is due. This radiating heat therefore has little or no effect in heat-
ing the atmosphere during its transmission to the Earth's surface ;
but after falling upon and heating terrestrial objects, it loses the
power of radiating completely through the atmosphere, and is trans-
mitted back into space through the atmosphere by conduction, con-
vection, and partial radiation to limited distances. But for any of
these modes of transmission, it is essential that the temperature of
the atmosphere should be greater in its lower than in its upper por-
tions, and in a degree greater as the quantity of heat to be trans-
mitted is greater. The temperature {t^ of the upper portion must
be determined by the condition, that, in a given time, a quantity of
heat must radiate from it into surrounding space equal to that which
falls upon it from external sources in the same time, and is trans-
mitted back after reaching the surface of the earth or objects near
to it. Consequently t^ must be independent of the height of the
Earth's atmosphere. At lower points the temperature will increase
till we reach the surface of the Earth ; and if we denote the tempe-
rature there by Tj, it is manifest that r, will be greater, the greater
the height of the Earth's atmosphere.
It must here be particularly observed, that t^ is the proper tern-
151
perature of the component particles of the atmosphere, and is pro-
bably widely different from the temperature which would be indi-
cated by a thermometer placed at the upper extremity of the atmo-
sphere, since the instrument would not only be affected by the ex-
change of heat between its bulb and the atmospheric particles, but
also by the heat radiating upon its bulb from every source of heat in
surrounding space; while the atmosphere, on account of its diather-
mancy, would remain unaffected by this radiating heat.
Conceive now a thermometer to be placed at a point sufficiently
above the earth's atmosphere. If the bulb were sheltered from the
direct influence of the solar rays, the thermometer would indicate
the temperature of that point of space, independent of the effect of
radiation from the central luminary of the solar system, but depen-
dent on the radiation from all other sources of heat in the universe.
If the instrument thus sheltered were sufficiently remote from the
sun and every planet, it would indicate very nearly the same tem-
perature at every point within the solar system, assuming the absence
of all unknown centres of heat within that system or near to it.
This is what may be understood by the general temperature of pla-
netary space. Let it be denoted by T. We shall then have T greater
than To ; and therefore if we now conceive the thermometer to be
transported to the upper limit of the atmosphere, it will be affected
by the lower temperature there, and will indicate a temperature in-
termediate to T and 73. If the instrument be brought still lower
within the atmosphere, it .will indicate a still lower temperature,
from its being entirely surrounded by a portion of the atmosphere
more dense than that at the extreme boundary, till this tendency to
lower the indications of the thermometer is counteracted by the
greater temperature of the atmospheric particles as we descend
towards the Earth's surface. At some point, consequently, within
the Earth's atmosphere the indication of the thermometer would
attain its minimum ; after which, in descending continuously towards
the Earth, the temperature indicated would constantly increase,
omitting variations due to temporary or local causes. Thus it fol-
lows that the ^istence of an atmosphere like that of the Earth, en-
veloping a planet, may, according to its extent, either elevate the
superficial temperature of the planet above, or depress it below that
of surrounding space independently of the direct solar radiation.
With respect to our own globe, we are entirely ignorant of the height
to which the thermometer, in ascending, would continue to indicate
a decreasing temperature, but we are sure that such height is great.
This is important with reference to the ultimate object of this paper ;
for if the height of a planet's atmosphere were too small to allow a
thermometer descending in it to attain its minimum indication, it is
manifest that an increase of atmosphere would cause a decrease in the
planet's superficial temperature ; whereas if the height of the atmo-
sphere were great enough to allow the thermometer to attain the mini-
mum, diuy increase of atmosphere would necessarily cause an increase in
the superficial temperature of the planet. In the Earth's atmosphere,
we are sure (as just remarked) that the indications of a thermometer
152
would constantly increase in its descent from a very high point above
the Earth's surface ; and therefore it follows, that if a planet be en-
veloped in an atmosphere similar to that of the Earth, but of greater
height, the superficial temperature of that planet will be higher than
that of the Earth, supposing both to exist in the planetary space un-
affected by the heat which radiates from the Sun ; while, on the
contrary, the superficial temperature of the planet would necessarily
be less, under the same conditions, than that of the Earth, if its
atmosphere were smaller, unless it should be so small as not to allow
a thermometer descending in it to reach its minimum indication. If
the planet were entirely without atmosphere, its superficial tempe-
rature (in the assumed absence of solar radiation) would be that of
surrounding space ; but we have no means of determining what rela-
tion that temperature bears to existing terrestrial temperature, or to
what this latter temperature would become in the absence of solar
radiation.
The author has calculated from Poisson's formulae the increase of
temperature in the superficial crust of the Earth, due to the amount
of heat received by direct radiation from the Sun, in diiFerent lati-
tudes, above that temperature which would be common to all parts
of the Earth's surface in the absence of solar radiation, and with a
uniformity of intensity of stellar radiation in all directions upon our
globe. But -this increased temperature must produce an augmentation
of temperature in the atmosphere, which must react on the terrestrial
temperature till equilibrium of temperature be established. The
author has endeavoured to estimate the amount of this indirect eflfect
of solar radiation by means of the data furnished by M. Dove's work
on terrestrial temperatures, combined with calculations based on
Poisson's formulae. He concludes that the whole effect of solar heat
at any proposed place is very nearly double that due to the im-
mediate and direct effect of solar radiation. Having thus ascertained
this entire effect, he finds the temperature which would pervade the
whole surface of the earth if the solar heat were extinguished. He
estimates this temperature at — 39°*5 C.
The annual variation of temperature in any latitude is found to be
nearly the same in amount for the terrestrial surface and for the part
of the atmosphere resting upon it. This must be understood as
applying to those places at which the temperature is not materially
affected by the horizontal transference of heat by marine or aerial
currents, or any local causes, which disturb the dependence of tem-
perature on latitude alone. The author also points out the depend-
ence of the annual inequalities of the terrestrial temperature (and
consequently of those also of the atmosphere) on the conductivity
and specific heat of the matter which constitutes the Earth's crust.
If these were much greater, the annual changes of temperature would
be much less.
Before applying these results to other planets, the author states
that he does not admit the notion, that the remoter planets may derive
a considerable superficial temperature from the remains of that in-
ternal heat which they probably possessed in the earher stages of
153
their existence. It is a well-established conclusion, that the super-
ficial temperature of our own globe has arrived at that point below
which it can never descend by more than the small fraction of a
degree, so long as all external conditions remain the same as at
present ; and the superficial temperature of the remoter planets will
in all probability be reduced to the corresponding limit. To these
external conditions, therefore, and not to their primitive heat, must
the existing temperatures on the surfaces of these planets be attri-
buted, assuming always that they are not of less antiquity than our
own globe. Hence the superficial temperature of the Earth, with its
present atmosphere, placed at the distance of Neptune, Uranus, or
Saturn, would be very nearly — 39°'5 C, since the effect of solar
radiation at those distances would be nearly insensible. But if the
extent of the atmosphere were increased, the superficial temperature
would be augmented in a corresponding degree. Judging by the
decrements of temperature observed by Mr. Welsh, the author con-
cludes that an increase in the height of the Earth's atmosphere of
35,000 or 40,000 feet, would elevate her superficial temperature, if
placed in the remoter planetary regions, to nearly the mean tempe-
rature of our present temperate zone. The same conclusion will
hold with respect to the three planets above mentioned, if we sup-
pose them to have atmospheres similar to that of the Earth, and of
sufficient extent. Their temperatures must be sensibly uniform over
the whole of their surfaces, not being subject to any appreciable
annual variation.
The same conclusions will apply to Jupiter, except that there will
be a small augmentation of temperature arising from solar radiation,
which the author calculates might amount to about 2^° C. at his
equator.
Hence the author concludes that those views which assign a
necessarily low temperature to the above-mentioned planets in con-
sequence of their distance from the Sun, are altogether untenable.
The conditions under which Mars is placed approximate more
nearly to those of the Earth than for any other planet. The author
calculates, that with an atmosphere similar to that of the Earth, and
exceeding it in height by about 15,000 or 20,000 feet, the equatorial
temperature of Mars may be about 60° F., or 15°'5 C, and his polar
temperature about — 10°C. The extent of the annual variations
would be about half those on our own planet in corresponding lati-
tudes, supposing the conductivity, specific heat, and radiating power
of the matter composing his superficial crust to be the same as for
the Earth.
Again, if the Earth, with her present atmosphere and obliquity,
were placed in the orbit of Venus, the mean equatorial temperatiire
would be upwards of 90° C, subject to the reduction, which would
doubtless in this case be great, due to the horizontal transference of
heat. The mean polar temperature would be about 16° C. A
diminution in the atmosphere would reduce these temperatures in
any assigned degree. But the obliquity of Venus, though not satis-
factorily determined, is considered to be much greater than that oi
154
the Earth, amounting, according to the estimate of some astronomers,
to as much as 75°, as heretofore stated. This would of course
render the character of her seasons entirely different from those of
the Earth. The greatest mean annual temperature would be at the
pole. Independently of the horizontal transference of heat by aerial
currents or other causes, taking the extreme obliquity of 75°, and sup-
posing the atmosphere of Venus to be exactly like that of the Earth,
her mean temperature at the equator would be about 56° C, and at
the pole 95° C. This latter would probably be much lowered by
currents ; but if the height of the atmosphere of Venus be less than
that of the Earth's atmosphere by about 25,000 feet, the author con-
siders that the mean temperature of Venus in her equatorial regions
would not exceed that of the temperate regions of the Earth ; while
the mean polar temperature would probably be about 40° C, or
about 12° or 13° C. higher than the Earth's equatorial temperature.
The heat of sunshine may be moderated by an atmosphere more laden
with vapour than that of the Earth.
Supposing the atmosphere of Venus like that of the Earth in its
nature and its magnitude, the temperature at her poles, with the
supposed obliquity, must be subject to an enormous awraMaHnequality,
amounting to between 70° and 80° C. above or below the mean tem-
l)erature, liable, however, to a great reduction by horizontal trans-
ference of heat. It may also be considerably reduced by the nature
of the matter which constitutes her outer crust. A reduction, like-
wise, in the extent of her atmosphere, like that above supposed,
would probably diminish the amount of this inequality, as well as
the mean temperature, though not in the same degree. It is easy
to conceive that the coefficient of the inequality may be thus reduced
to some 40° C. ; and supposing the mean temperature then, as above
estimated, at about 40° C, the annual polar temperature will oscil-
late between 0° C. and 80° C. At the equator, the semi-annual in-
equality might amount, under the above suppositions, to about 10°
or 12° C, in which case the equatorial temperature might oscillate
between something below zero (C.) and some 25° C. It should be
recollected also, that a much greater reduction of the mean tempera-
ture would result from a greater reduction in the extent of this
planet's atmosphere than above supposed with reference to the height
of our own atmosphere. This would not, the author conceives, be
inconsistent with the existence of a large quantity of vapour in the
atmosphere, affording shelter from the heat and glare of sunshine.
The Moon is under the peculiar circumstances of the absence
of a sensible atmosphere, and her long period of rotation about her
axis. Assuming her to have no atmosphere at all, the mean tem-
perature of her outer crust, in the absence of the Sun, would be the
general temperature of that portion of planetary space in which the
solar system is situated. How much this might differ from the
superficial temperature which the Earth would have with the like
absence of the Sun, and which the author estimates at — 39°*5 C,
as above stated, it is impossible to determine ; but whatever it may
be, the influence of the Sun's heat would be to increase it by about
155
40° C. at the Moon's equator, and by a small amount only at her
poles. This must be attended by an enormous monthly inequality,
amounting to nearly 60° C, supposing the matter of which her su-
perficial crust is composed to have the same conductivity, specific
heat, and radiating power as the crust of the Earth. If these be
much greater for the Moon, this inequality might be considerably
diminished. At the poles it must be comparatively small.
The lunar temperatures here spoken of are those of the matter
forming her external crust. The temperature which would be indi-
cated by a thermometer placed in her immediate vicinity would be
aflFected by the Moon (in the assumed absence of an atmosphere)
only by her direct radiation. We have not the means of determining
what this temperature may be.
Also a paper was read " On the singular Points of Curves." By
Professor De Morgan.
Mr. De Morgan defines a curve as the collection of all points
whose co-ordinates satisfy a given equation ; and contends for this
definition as necessary in geometrical algebra, whatever limitation
may be imposed in algebraic geometry. He divides singular points
into points of singular position and points of singular curvature ; the
character of the former depending on the axes, but not that of the
latter. Both species are defined as possessing a notable property,
and such as no arc of the curve, however small, can have at all its
points.
The form first considered is that of which the case usually taken
is an algebraic curve. Let (p(x, y) be a function which for all real
and finite values of x and y is real, finite, and univocal ; let the curve
be 4 + a?"y— a?sy2 ^ 2x7y3 _a^4y4 _|_y6 _3^,y3 .{.^jJ^y's =0,
and discusses its infinite branches, and the sextuple point at the
origin (which turns out to be a couple of isolated points, and a cusp
of similar flexures), with very much less space and trouble than ordi-
nary methods would demand from a much less complicated instance.
It is also shown that the lower form of Lagrange's theorem solves the
following question : — Given an equation with a certain number of
equal roots, what effect will be produced uj)on these roots by given
infinitesimal alterations in the coefficients, how many will remain
real, and how many will become imaginary ?
Newton has given the foundation and the chief step of a geome-
* There attributed to Mr. Minding, by a mistake caused by M. Serret,
who incorporates it with a theorem of Mr. Minding, without any notice of
its author.
157
trical method (Newton s parallelogram) which, has passed into oblivion,
though it occurs in the celebrated second letter to Oldenburg, has
been fully described by Stirling, used by Taylor and De Gua, and
forms the main method of Cramer's work on curves. Mr. De Mor-
gan proposes to call it the method of co-ordinated exponents.
He proceeds to describe and enlarge this method ; observing that,
of the polygon which represents an equation, Newton and his fol-
lowers are in full possession of the connexion of the sides with the
solutions, and fail only in not grasping the connexion of the whole
polygon with the whole equation. Both Newton's method and
Lagrange's, the second of which is an arithmeticeJ version of the
first, may be applied to irrational equations, but it will be convenient
to confine the description to the form S(w;'"y'»^0, where m and n
are integers.
In ax"']/", let n be an abscissa, and tn an ordinate, and let (m, n)
be called the exponent point of the term ax"^y^. Take some paper
ruled in squares (or ruled both ways in any manner, for any equal
rectangles will do) to facilitate the process when n and m are always
integers, and lay down all the exponent points in Saa:'"y"=0.
Through some of these points draw a convex polygon including all
the rest, which can only be done in one way. Should the points be
so many and so scattered that some method must be applied, the
geometrical method is a translation of the main arithmetical method
of Lagrange's theorem. The points which end on, or otherwise
fall in, the sides of the polygon show the essential terms of the equa-
tion : no others are wanted to determine q and u in i/=x^(u-\-U).
The upper contour of the polygon shows how all the solutions com-
mence in descending powers of x ; the under contour does the same
for ascending powers. Take any side of either contour, its projec-
tion on the axis of n shows the number of roots it represents, the
tangent of the angle it makes with the negative side of the axis of n
shows the value of r.
It will not be needful to abstract the developments given in the
paper : we shall only notice the inverse method. Tlie following
example is taken, and the construction of the equation is even easier
(under Cramer's form) than the direct treatment of it. The example
chosen by the author is the following : — Required ^(j;, y) = 0, of the
twelfth dimension in terms of y, such that the twelve roots of y,
with reference to lower degrees, shall be as follows : two roots of
the degree 1, four of ^, two of 0, one of — 1, two of — f , one of —2.
But with reference to higher degrees, there are to be one root of the
degree 3, two of \, three of 0, three of —\, two of —1, one of —3.
On examination these conditions are found compatible, and the most
general equation which satisfies the conditions is found.
The paper is terminated by a discussion on the pointed branch,
for the admission of which, as a branch altogether composed of sin-
gular points, the author contends.
158
November 12, 1855.
A paper was read by the Master of Trinity on the Intellectual
Powers according to Plato.
Also, Prof. Sedgwick gave a lecture on the Classification and
Nomenclature of the Palaeozoic Rocks.
November 26, 1855.
A paper was read on the Earthquake in Switzerland in July last,
by the Rev. O. Fisher.
The 25th of July, 1855, on which the first and most severe shock
was felt, was a very wet, close day, and the little wind stirring came
from the S.W.
In the Munster Thai the earthquake began by a rumbling vibra-
tion like that caused by a carriage run under an archway, gradually
increased for about four seconds, and then suddenly ceased. The
oscillation seemed to be from E.S.E. to W.N.W., but would be
aiFected by the build of the house.
In the church at Bienne two stones fell from the groining thirty
or forty feet into the organ pipes, to a point between 2 and 2^ feet
N. by E. of the point vertically beneath their first position ; and
allowing for the direction of the building, this would give the motion
of the earth about from N.E. to S.W. This wave may have been a
reflexion caused by the wave entering the Jura from the valley.
Another shock was felt at Bienne, at 10 a.m., on the 26th.
The great shock was felt at Strasburg, slightly at Lyons in a
direction from E. to W. ; likewise at Chambery, Alessandria, and
Genoa. The account given by Plana in the Times does not seem
very intelligible, but as far as can be made out from the stopping of
the clocks, it gives the direction of the shock at Turin about 30° W.
of S. Chiavenna, the western shore of the Lake of Constance, and
Schaflfhausen seem to fix the limit to which it was felt towards the
east. The area shaken was therefore an oval, having its largest
dimension about 300 miles N. and S., and its shortest 250 miles E.
and W.
At Geneva the shock appeared to be directed to E.N.E. At Thun
it appeared to come from Frutigen. At Kandersteg, at the north
foot of the Gemmi, the shock was N. and S. At Interlaken the
shocks were more severe ; and at Orroont, Canton Vaud, the oscil-
lation came from W. to E., preceded by a noise which lasted for an
instant only, and the roof of a house fell in. It seems that nearer
the centre of the oval the intensity of the shock was greater. At
the baths of Leuk a chimney was thrown down and the walls cracked ;
but on ascending the valley of the Rhone the evidence of disturbance
became rapidly more marked up to Visp, where only seven houses
remained habitable. At the little inn, the " Soleil," the flag pave-
159
ment was burst upwards as if by a blow from beneath : a continual
succession of shocks have occurred there at variable intervals up to
the present time. Passing on towards Brieg, the evidence of the
violence of the shock rapidly diminished. The valley of Zermatt
showed the chief disturbance ; the bridle road was continually fis-
sured, and in some places slipped down into the veJley. At Stalden
there was much destruction, but at St. Nicholas the havoc was very
great indeed. Higher up the traces of the shock were less and less,
until at Tesch, Randa, and Zermatt, there was no mischief done.
The other branch of the valley by Saas did not suffer so much.
Drawing lines through the different places in the direction in
which the wave proceeded, it will be found that they converge very
nearly to Visp, showing that to be nearly the centre of disturbance.
Mr. Croker of Caius College was walking between Stalden and
Visp when the great shock occurred, which appeared to him to be a
blow from beneath like the springing of a mine under him, and he
observed that the path sunk several inches from the solid rock ; a
lofty isolated rock on the opposite side of the valley vibrated, and
blocks of stone came tumbling down on all sides. The quivering
lasted about thirty seconds. He did not observe any sound prece-
ding the shock, though this was heard at Visp ; but a crashing
sound accompanied the great shock, and a fainter sound continued
afterwards beyond the motion. He felt continued shocks from one
o'clock till four, when he proceeded towards Sion. At Zermatt the
same shock was felt very much less violently, and no sound preceded
it ; and after attaining its maximum, it ceased somewhat suddenly.
It was felt less strongly on the RifFelberg; and on the 27th another,
felt at Zermatt as strongly as before, was not felt on the little Mont
Cervin .
A sound seems in general to have preceded the earthquake at
places near the centre of disturbance : at Visp likened to the echo
of an avalanche, but at a distance there was only a sound simul-
taneous with the shock. The sound may have arisen from the grind-
ing of the walls of the fissure, or whatever violent action may have
occurred at the origin, and the sound-waves travelling more rapidly
than the earthquake- wave. This is opposed to Mr. Mallet's view,
though he gives a table in which the least rate given for sound tra-
velling through any kind of stone is 3640 feet per second, while the
rate of motion of the earthquake of Lisbon was 1750 feet. If the
view stated be correct, the disturbance must have been deep in the
earth, which would also explain the upward blow felt by Mr. Croker.
At greater distances the sound-wave would be expended sooner than
the earthquake-wave, and the accompanying sound be due to local
action.
Chimneys and such like structures appear to have fallen away
from the centre of disturbance, being thrown down by the return
stroke of the wave ; the forward stroke having to move them only
from a state of rest, whereas the return stroke would have to over-
come the momentum generated by the former.
Near the centre the shock was sudden, passing away gradually.
160
At a distance it began with slight quivering, gradually attained a
maximum, and then suddenly ceased. Now if the disturbance oc-
curred along a large fissure, perhaps several miles in length, and of
unknown depth, the waves from different portions would reach any-
given point in succession, and at intervals the combined effect of
many waves would be felt, producing a result analogous to the rolling
of thunder due to the varying distance of the source of sound, while
the sudden concussion at a nearer point is like the detonation heard
when the lightning is near the auditor.
The shocks were less severe in the mountains than in the valleys.
As far as the wave progressing horizontally is concerned, it would,
on entering a mountain, at first be nearly bounded by a horizontal
plane continuous with that of the valley, just as light is propagated
in straight lines ; but there would also be a diversion of a part (ana-
logous to the diffraction of light at a screen) into the mountain, so
that where the wave passed for some distance into a range it would
finally be felt at the summit. It is observable that the shock on the
25th was less severe on the Riffelberg than at Zermatt, yet it travelled
through the mountain and was felt at Turin.
The period of elevation of the Alps seems about contemporaneous
with the older Pliocene of Sir C. Lyell. The country is broken up
with faults, which probably there, as elsewhere, follow the lines of
valleys. The valley of Visp lies in the axis of two ranges which
have all the appearance of a mighty valley of elevation. The shock
may have arisen from a shifting of the beds on this line of ancient
disturbance, and very probably the somewhat rectangular corner
between the valleys at Visp suffered the principal displacement.
Earthquakes in non-volcanic regions probably arise from a failure of
support. At the period of the elevation of the Alps, the more heated
lower parts of the earth's crust must have come nearer to the surface
than their normal position, and contractions and failure of support
must occur while cooling, and the comparatively recent elevation of
the Alps may give reason for thinking this to be still going on.
December 10, 1855.
t
A paper was read by Mr. Maxwell on Faraday's Lines of Force.
The method pursued in this paper is a modification of that mode
of viewing electrical phjenomena in relation to the theory of the uni-
form conduction of heat, which was first pointed out by Professor W.
Thomson in the Cambridge and Dublin Mathematical Journal, vol. iii.
Instead of using the analogy of heat, a fluid, the properties of which
are entirely at our disposal, is assumed as the vehicle of mathematical
reasoning. A method is given by which two series of surfaces may
be drawn in the fluid so as to define its motion completely. The
uniform motion of an imponderable and incompressible fluid permea-
ting a medium, whose resistance is directly as the velocity, is then
discussed, and it is shown how a system of surfaces of equal pressure
161
naay be drawn, which, with the two former systems of surfaces,
divides the medium into cells, in each of which the same amount of
work is done in overcoming resistance. It is then shown that if the
fluid be supposed to emanate from certain centres, and to be absorbed
at others, the position of these centres can be found when the pres-
sure at any point is known ; and that when the centres are known,
the distribution of pressures may be found. Methods are then given
by which the motion of the fluid out of one medium into another,
the resistance of which is different, may be conceived and calculated,
and the theory of motion in a medium in which the resistance is
difi'erent in different directions is stated.
The mathematical ideas obtained from the fluid are then applied
to various parts of electrical science. It is shown that the expres-
sion for the electrical potential at any point is identical with that of
the pressure in the fluid, provided that " sources " of fluid are put
instead of positive electrical " matter," and centres of absorption or
" sinks " for negative " matter."
The theory of Faraday with respect to the effect of dielectrics in
modifying electric induction, is illustrated by the case of difi'erent
media having different conducting power ; and it is shown, that, in
order to calculate the effects by the ordinary formulae of attractions,
we must alter in a certain proportion the quantities of electricity
within the dielectric, and conceive an imaginary distribution of elec-
tricity over the surface which separates it from the surrounding
medium.
The theory of magnets and of the phaenomena of paramagnetic
and diamagnetic bodies is expressed with reference to the " lines of
inductive magnetic action;" and elementary proofs of the tendency
of paramagnetic bodies toward places of stronger magnetic action,
and of diamagnetic bodies toward places of weaker action, are given.
This distinction of paramagnetic and diamagnetic is not here used
absolutely, but indicates a greater or less conductivity for the lines
of inductive action than that of the surrounding medium.
The magnetic phaenomena of crystals are then examined, and
referred to unequal magnetic conductivity in different directions ;
and the case of a crystalline sphere in a uniform field of force is
worked out.
The laws of electric conduction, as laid down by Ohm, are shown
to agree with those of the imaginary fluid, and definitions of quan-
tity and intensity are given, which will apply to magnetism as well
as galvanism.
The theory of the attractions of closed circuits, as established by
Ampere, is shown to lead to the following results : —
1. The total intensity of the magnetizing force estimated along
any closed curve embracing the circuit is a measure of the quantity
of the current.
2. The quantity of the current, multiplied by the quantity of in-
ductive magnetic action, from whatever source, which passes through
it, gives what may be called the potential of the circuit. The ten-
dency of the resultant forces is to increase this potential.
162
The theory of Faraday with respect to the induction of currents in
closed circuits takes the following form : —
When the quantity of inductive magnetic action which passes
through a given circuit changes in any way, an electromotive force
proportional to the rate of change acts in the circuit, and a current
is produced whose quantity is the electromotive force divided by the
total resistance of the circuit.
The mathematical discussion of the electro-magnetic laws is re-
served for another communication.
PROCEEDINGS
CAMBRIDGE PHILOSOPHICAL SOCIETY.
February 11, 1856.
A paper was read by Mr. Maxwell of Trinity College, " On Fara-
day's Lines of Force," in continuation of a former paper (Proceedings
of the Society, Dec. 10, 1855).
This paper was chiefly occupied with the extension of the method
of lines of force to the phaenomena of electro-magnetism, by means
of a mathematical method founded on Faraday's idea of an " electro-
tonic state."
In order to obtain a clear view of the phaenomena to be explained,
we must begin with some general definitions of quantity and inten-
sity as applicable to electric currents and to magnetic induction. It
was shown in the first part of this paper, that electrical and mag-
netic phaenomena present a mathematical analogy to the case of a
fluid whose steady motion is affected by certain moving forces and
resistances. fThe purely imaginary nature of this fluid has been
already insisted on.] Now the amount of fluid passing through any
area in unit of time measures the quantity of action over this area ;
and the moving forces which act on any element in order to over-
come the resistance, represent the total intensity of action within the
element.
In electric currents, the quantity of the current in any given direc-
tion is measured by the quantity of electricity which crosses a unit
area perpendicularly to this direction ; and the intensity, by the
resolved part of the whole electromotive forces acting in that direc-
tion. In a closed circuit, whose length =^, coeflicientof resistance
=k, and section =0^, if F be the whole electromotive force round
the circuit, and I the whole quantity of the current,
C2 Ik
The laws of Ohm with respect to electric currents were then ap-
plied to cases in which the conducting power of the medium is dif-
ferent in difi'erent directions. The general equations were given and
several cases solved.
In magnetic phaenomena, the distinction of quantity and intensity
is less obvious, though equally necessary. It is found, that what
Faraday calls the quantity of inductive magnetic action over any
No. XII. — Proceedings of the Cambridge Phil. See.
164
surface, depends only on the number of lines of magnetic force which
pass through it, and that the total electromotive elFect on a conduct-
ing wire will always be the same, provided it moves across the same
system of lines, in whatever manner it does so. But though the
quantity of magnetic action over a given area depends only on the
number of lines which cross it, the intensity depends on the force
required to keep up the magnetism at that part of the medium ; and
this will be measured by the product of the quantity of magnetiza-
tion, multiplied by the coefficient of resistance to magnetic induction
in that direction.
The equations which connect magnetic quantity and intensity are
similar in form to those which were given for electric currents, and
from them the laws of diamagnetic and magnecrystallic induction
may be deduced and reduced to calculation.
We have next to consider the mutual action of magnets and elec-
tric currents. It follows from the laws of Ampere, that when a
magnetic pole is in presence of a closed electric circuit, their mutual
action will be the same as if a magnetized shell of given intensity
had been in the place of the circuit and been bounded by it. From
this it may be proved, that (1) the potential of a magnetized body on
an electric circuit is measured by the number of lines of magnetic
force due to the magnet which pass through the circuit. (2) That
the total amount of work done on a unit magnetic pole during its
passage round a closed curve embracing the circuit depends only on
the quantity of the current, and not on the form of the path of the
pole, or the nature or form of the conducting wire.
The first of these laws enables us to find the forces acting on an
electric circuit in the magnetic field. Give the circuit any displace-
ment, either of translation, rotation, or disfigurement, then the dif-
ference of potential before and after displacement will represent the
force urging the conductor in the direction of displacement. The
force acting on any element of a conductor will be perpendicular to
the plane of the current and the lines of magnetic force, and will be
measured by the product of the quantities of electric and magnetic
action into the sine of the angle between the direction of the electric
and magnetic lines of force.
The second law enables us to determine the quantity and direction
of the electric currents in any given magnetic field ; for, in order to
discover the quantity of electricity flowing through any closed curve,
we have only to estimate the work done on a magnetic pole in passing
round it. This leads to the following relations between a^ jSi y^,
the components of magnetic intensity, and Og ^2 ^2' ^^^ resolved parts
of the electric current at any point,
a — ^—ttL b =^——' c„=-^-^.
dz dy dx dz' ' dy da;
In this way the electric currents, if any exist, may be found when
we know the magnetic state of the field. When a^dw+fi^dy+y^dz
is a perfect differential, there will be no electric currents.
Since it is the intensity of the magnetic action which is immedi-
165
ately connected with the quantity of electric currents, it follows that
the presence of paramagnetic bodies, like iron, will, by diminishing
the total resistance to magnetic induction while the total intensity
is constant, increase its quantity. Hence the increase of external
effect due to the introduction of a core of soft iron into an electric
helix.
From the researches of Faraday into the induction of electric cur-
rents by changes in the magnetic field, it appears that a conductor,
in cutting the lines of magnetic force, experiences an electromotive
force, tending to produce a current perpendicular to the lines of
motion and of magnetic force, and depending on the number of lines
cut by the conductor in its motion.
It follows that the total electromotive force in a closed circuit is
measured by the rate of change of the number of lines of magnetic
force which pass through it ; and it is indifferent whether this change
arises from a motion of this circuit, or from any change in the mag-
netic field itself, due to changes of intensity or position of magnets
or electric currents.
This law, though it is sufficiently simple and general to render
intelligible all the phsenomena of induction in closed circuits, con-
tains the somewhat artificial conception of the number of lines pass-
ing through the circuit, exerting a physical influence on it. It
would be better if we could avoid, in the enunciation of the law,
making the electromotive force in a conductor depend upon lines of
force external to the conductor. Now the expressions which we
obtained for the connexion between magnetism and electric currents
supply us with the means of making the law of induced currents
depend on the state of the conductor itself.
We have seen that from certain expressions for magnetic intensity
we could deduce those for the quantity of currents, so that the cur-
rents which pass through a given closed curve may be measured by
the total magnetic intensity round that curve. Here we have an
integration round the curve itself mste^di of one over the enclosed sur-
face. In the same way, if we assume the mathematical existence of
a state, bearing the same relation to magnetic quantity that mag-
netic intensity bears to electric quantity, we shall have an expression
for the quantity of magnetic induction passing through a closed cir-
cuit in terms of quantities depending on the circuit itself, and not
on the enclosed space.
Let us therefore assume three functions oi xy z, ao/3oyo» such that
Ci bi Ci being the resolved parts of magnetic quantity,
^ dz dy ' ' dx dz ' dy dx '
then it will appear that if we assume -J^, -^, -J^ as the expres-
sions for the electromotive forces at any point in the conductor, the
total electromotive force in any circuit will be the same as that ex-
pressed by Faraday's law. Now as we know nothing of these in-
ductive effects except in closed circuits, these expressions, which are
166
true for closed currents, cannot be inconsistent with known phteno-
mena, and may possibly be the symbolic representative of a real law
of nature. Such a law was suspected by Faraday from the first,
although, for want of direct experimental evidence, he abandoned his
first conjecture of the existence of a new state or condition of matter.
As, however, we have now shown that this state, as described by
him (Exp. Res. (60.)), has at least a mathematical significance, we
shall use it in mathematical investigations, and we shall call the
three functions ao, fio, yo, the electrotonic functions (see Faraday's
Exp. Res. 60. 231. 242. 1114. 1661. 1729. 3172. 3269.).
That these functions are otherwise important may be shown from
the fact, that we can express the potential of any closed current by
the integral
J(«,s^+JA|+^.yo|)*>
and generally that of any system of currents in a conducting mass
by the integral
111 (jx^a^-\- (i(f}^ + yQC^)dx dy dz.
The method of employing these functions is exemplified in the
case of a hollow conducting sphere revolving in a uniform magnetic
field (see Faraday's Exp. Res. (160.)), and in that of a closed wire
in the neighbourhood of another in which a variable current is kept
up, and several general theorems relating to these functions are
proved.
February 25, 1856.
A paper was read, " On a direct method of estimating Velocities,
Accelerations, and all similar magnitudes with respect to Axes move-
able in any manner in Space, with applications." By Mr. Hayward,
of St. John's College.
The frequent recurrence, in many different investigations of kine-
matics and dynamics, of exactly corresponding equations, suggests
the inquiry whether they do not result from some common principle,
from which they may be deduced once for all. An investigation
based on this idea forms the first part of this paper, and the result
is the method mentioned in the title.
This calculus shows how the variations of any magnitude, capable
of representation by a straight line of definite length in a definite
direction, and subject to the parallelogrammic law of combination,
may be simply and directly determined relatively to any axes what-
ever. ^ If such a magnitude (m) be estimated in a given direction, its
intensity in that direction will be represented by the projection on
it of the line which represents u. If this given direction be not
fixed, but move according to a given law, the projection of u upon it
will change by the alteration of its inclination to the direction of u ;
and the rate of that change is easily calculated, whence an expres-
167
siou for the acceleration of the resolved part of a along a given axis
as due to the motion of that axis. If u itself be variable, its variations
may be conceived to be due to an acceleration / in a definite direc-
tion, which in the time dt produces a quantity /rf? in the direction
of / to be combined with u by the parallelogrammic law ; hence
result expressions for the changes in intensity and direction of u.
If, u being variable, the variations in its intensity estimated along a
given moveable direction be sought, it will consist of two parts : one,
that due to the resolved part of/ in the given direction; the other,
that due to the motion of the axis, wliich is the same as if/ had not
existed, or u had been constant : hence expressions for the total
acceleration of the resolved part of u along the given moveable axis.
If u be resolved along three rectangular axes, these expressions take
the forms of familiar kinematical and dynamical equations.
These results furnish immediately expressions for the relative
velocities of a point with respect to moving axes when its absolute
velocities in their directions are given, and vice versd. They also
furnish very ready means of estimating accelerations in variable di-
rections ; as, for instance, the radial and transversal accelerations of
a point moving in a plane or in space, or the tangential and normal
accelerations in the same case. These are some kinematical appli-
cations of the calculus.
The dynamical applications form the second part of the paper.
Here the general problem of the motion of a system, so far as it is
due to external forces, is divided into two steps : one ivova force to
momentum, the other from momentum to velocity. If the momenta of
the particles of a system be reduced like a system of forces, they
produce a single linear momentum and a single angular momentum,
just as a system of forces produces a single force and a single couple.
The linear momentum is (in our received language) the momentum
of the mass of the system collected at its centre of gravity ; the an-
gular momentum is a magnitude the constancy of whose intensity
in a given axis is equivalent to the assertion of the principle of the
conservation of areas for that axis, and the constancy of whose direc-
tions determines the " invariable plane " as a plane perpendicular to
it. The momentum, whether linear or angular, is a magnitude to
which the previous calculus applies, and the resultant force and
resultant couple are respectively the accelerators of the two kinds of
momentum : hence the equations obtained in the first part, inter-
preted with respect to these magnitudes, furnish equations in any
required form for the determination of the momenta at any instant.
The step from force to momentum is independent of the nature of
the system, that from momentum to velocity requires the system to be
particularized. In the paper the case of an invariable system only
is considered, and in particular its motion of rotation about its centre
of gravity. The axis of rotation or angular velocity is related in
direction to that of angular momentum, as the radius of the central
ellipsoid with which it coincides to the normal at its extremity.
Hence an angular momentum constant in intensity and direction,
in general gives rise to an angular velocity variable in both respects,
168
and vice versd. The question then becomes, to determine the acce-
leration of angular velocity due to the motion of the system. This
is obtained by determining the acceleration of angular momentum
for a line fixed in the body, which is then shown to be a maximum
for the normal to the plane containing the axes of angular momentum
and velocity ; then the acceleration along this line is the total acce-
leration of angular momentum due to the motion, and the accelera-
tion of angular velocity determined from it (just as the angular velo-
city is determined from the momentum) is that due to the motion
of the system. Also the acceleration of angular velocity due to the
forces is related to the resultant couple and its axis, just as the an-
gular velocity to the angular momentum. Thus the accelerations
of angular velocity due both to the motion and to the forces being
determined, the intensity and direction of the angular velocity at any
time is to be found by combining these eiFects by integration. The
problem is worked out in the case of the axis of the resultant couple
being coincident with that of angular momentum, so that this remains
fixed. The paper concludes with a simple solution of the problems
of Foucault's gyroscope as applied to show the efi^ects of the earth's
rotation, the simplk;ity arising from the method of this paper enabling
us at once to refer the motion to those axes (neither fixed in the
body nor in space) whose motion it is desired to determine.
March 10, 1856.
A paper was read by L. Barrett, Esq., " On the Distribution of the
Mollusca on the Coast of Norway." {Vide Annals of Nat. Hist.
May 1856.)
In this the author observed, that when the fauna of the coast of Nor-
way is compared with that of the other side of the North Atlantic, a
great difference will at once be perceived, not only in the number of
species, but also in the different distribution of northern and south-
ern types ; the Mollusca of Greenland being peculiarly arctic, those
of Scandinavia a mixture of southern and northern species. In the
southern part of Norway we find the species living on our coasts
abundant ; but they become rarer as we go north, their place being
supplied by arctic forms. Many of the northern species have a great
geographical range, at which we need not be surprised when we
consider their great antiquity, many of them having existed since the
pliocene period ; and, in the author's opinion, whenever we find a
species with a great geographical range, we may at once infer that
it has continued to live from a remote period. It is extremely diffi-
cult, according to the present state of the currents in the northern
seas, to account for the wide distribution of arctic shells on this side
of the North Atlantic ; but when we consider that at not a very
distant period the temperature and other conditions of this area
were totally diflPerent, that a cold climate prevailed, certainly accom-
panied by a current setting from the north (as is fully proved by the
169
fact that boulders are always found nearly south of the moun-
tain ranges from which they have been originally transported), and
that many of the shells are found fossil in the Sicilian tertiaries,
this wide distribution may be fully accounted for. As these frigid
conditions were gradually altering to more genial ones, those species
requiring a lower temperature would gradually die out, and only con-
tinue to exist in higher latitudes. The littoral and shallow water
species would be most affected by such an alteration of climate ;
and while the fauna of the littoral and laminarian zones would be
entirely changed, — the shells composing that fauna replaced by other
species, — those living in the deep sea would continue to exist, per-
haps at a greater depth, mingled with the species brought in with
the new physical conditions of the area. This we know to be the
case ; for while the northern littoral shells, such as Mya truncata,
&c., are found only fossil in Sicily, many of the deep-sea arctic species
that existed there when those fossils were aUve are still found living
in the deeper parts of the Mediterranean.
The same thing occurs on our coasts, where the arctic littoral
or_ shallow-water shells, as Astarte arctica, TeUina proximo, Natica
heUcoides, &c., which are found in shallow water on the Scandi-
navian or Greenland coasts, are now rare as deep-sea shells, and that
in the same area in which they were formerly abundant as shallow-
water species. Some species are capable of enduring great differ-
ences of cKmate, the Mytilus edulis being found as abundantly on
the coast of Greenland as on our own shores.
It is not difficult to account for the presence of the southern species
on the coast of Norway, as the Gulf-stream sets directly along the
coast, warming its waters, and rendering them habitable for species
requiring a moderately high temperature. The great abundance and
wide distribution of these species show that the present order of
things has continued for a great length of time. The gradual ex-
tinction of northern shells on our coasts is still going on ; the
number of living specimens of Pecten danicus is very small, while
dead shells are very abundant, and fresh dead specimens of Pecten
islandicus are frequently dredged, though a living specimen has not
yet occurred. It is probable that this species has died out very
recently.
On the eastern shores of Davis's Straits the Mollusca are about
half as numerous as on the coast of Norway. The fauna differs in
the prevalence of arctic types and the total absence of southern. At
a former period the fauna was of a mixed character ; species now in-
habiting more southern latitudes are found fossil in the raised beaches
at Disco Island, which species are no longer found living on the coasts.
April 28, 1856.
A paper was read " On the Theory of Heat," by Mr. A. A. Har-
rison, of Trinity College.
The object of this paper was to show that there is considerable
170
reason for supposing that radiant heat is identical with light, and that
they both consist of vibrations of the ultimate particles of matter.
There is a strong presumption of this from the facts, that every
body heated to a certain temperature, dependent only on the nature
of the surface, emits light as well as heat ; and that " whenever light
manifests itself, heat appears along with it " (Kelland) : the difference
between radiant heat and ordinary heat is, that radiant heat is due
to vibrations in planes normal to its direction of propagation, and
that ordinary heat consists of vibrations in all three dimensions.
The author endeavoured to show, in the first place, that the mo-
tions of the particles of matter, which must be caused by friction, or
in the union of two gases in combustion, is sufficient of itself to
account for the following phaenomena of heat : —
I. That a body once heated continues of the same temperature,
with the exception of heat lost by radiation, conduction, &c. This
follows immediately from the principle, that in any system of par-
ticles held together by mutual attractions and repulsions, the vis viva
is independent of the time, and depends merely on the position of
the particles.
II. That bodies expand by heat.
Before proceeding to this, the author argued that in gases the
repulsive force varies inversely as the cube, and not, as usually stated,
the simple power of the distance ; that it is not true, without some
limitation, that the force varies as the inverse first power, was urged
from the fact that such a force would decrease more slowly than one
varying as the inverse square, and consequently would be the force
observed in astronomical phaenomena ; and even the oxygen of the
ocean would repel that of the air instead of attracting it. That the
force varies as the inverse cube was deduced from the law of elas-
ticity, that the density varies as the pressure ; for if a particle repels
other particles with a force varying as the inverse cube, it repels a
fixed plane of them with a force varying as the inverse first power.
That this is the case may be seen, by considering that though the
particle repels particles similarly situated with a force varying as the
inverse cube, yet the number of such particles varies directly as the
square of the distance, and therefore the whole eflfect upon the plane
varies inversely as the first power. And if this is true for a plane,
it is also true for the solid side of the containing vessel ; for any
solid may be considered as made up of a succession of planes.
The law being the inverse cube, it follows that in any position the
sum of the forces exerted by any particle on two particles, one on
each side of it, is least when that particle is half-way between them,
and increases the further the particle is removed from the middle
point. This is seen directly, for the value of 1^ i t^ is
(a + xy^ (a—xy
least when x=0. and increases until x=a. And therefore, in order
to produce the same force, it would be necessary that the mean
distance should be increased ; and hence if the particles of any aeri-
form body be in motion, the force exerted by them would be greater
than when at rest; that is, if the pressure to be supported be con-
171
stant, the average distance of the particles must increase, and the
body must expand.
III. That every aeriform body not in contact with a liquid expands
in the same proportion. This was accounted for by the circumstance,
that the increase of pressure depends only upon the ratio of the dis-
turbance to the original distance, and not at all upon the absolute
distance.
IV. That air and elastic fluids give out heat on compression. By
compression the absolute distance of the particles from one another
is diminished ; but the absolute motion remaining the same, the rela-
tive motion is increased.
V. That the same amount of heat is generated in two gases sub-
jected to the same pressure ; for the absolute distance of the particle
in both being diminished in the same proportion, and the absolute
motion remaining unaltered, the relative motion is increased in the
same proportion in both.
VI. The specific heats are inversely as the atomic weights. Here
it was necessary to show that mass is not necessarily proportional to
the quantity of matter, as usually stated ; or rather, that a body may
have a different mass when considered with regard to the molecular
force from what it has with respect to the force of gravity. "With
regard to elasticity of gases, the weight of any single particle is so
small as not to affect the result. The question remains, whether
we know anything of the masses of different particles relatively to
this repulsive force. To determine their masses we have these data.
In several different gases equivalent volumes under the same pressure
occupy the same space, that is (assuming the Daltonian theory, that
equivalent volumes contain the same number of particles), that each
particle of the two different gases exerts an equal pressure on the
adjacent particles : and hence, with reference to this law, the mass
of a single particle in each of these two different gases is the same,
and therefore the " vis viva " of equivalent weight or volumes subject
to the same motion is the same for both ; that is, the quantity of
heat of an equivalent of each is the same, and therefore the specific
heat of a given weight is inversely as its equivalent number or atomic
weight.
VVith reference to the phsenomena of radiation, it may be shown
from theoretical considerations that the inverse cube is the law
required. The inverse first is impossible, for then there could be no
vibrations. For the same reason the inverse second is impossible
(Camb. Phil. Trans. Vol. vii. p. 98). The inverse fourth is also im-
possible, for then there could be no vibrations, and the velocity
would be infinite (vol. vi. p. 325). It has also been shown that
neither the second nor the fourth would satisfy the conditions of the
equations (vol. vii. p. 419). Hence, from the theory of radiation,
it is supposed that the luminiferous aether consists of solid particles,
attracting one another with a force varying as the inverse square
(vol. vii. p. 110), and repelling with a force varying as the inverse
cube.
Now from the Daltonian theory, and the law of elastic fluids, it
172
has been shown that the ultimate particles of our atmosphere com-
pose such an aether. But if our atmosphere is the luminiferous aether,
we must next inquire whether it does pervade space. Omitting
variations of temperature, and merely considering the atmosphere as
subject to the two forces of elasticity and gravity, we have for the
equation of a column of air on a unit of surface,
dp=-9^pdz. or 1^=^, where ^=t
^ z^ p dz z^ p
Integrating this, we find that;) and p, though they become extremely
small, never vanish ; and therefore, if these laws are absolutely true,
our atmosphere does pervade space.
It may be well to obviate the objection, that black substances
radiate heat best, and white substances light. This arises from
employing the same word radiation to denote two different things :
by radiated heat is meant heat given out from a heated body ; by
radiated light is meant the secondary radiation from the surface of
a body exposed to light.
If Sir J. Leslie's experimental calculation of the heat lost from
the sun be correct, there is no need of any theories to account for its
generation.
From the foregoing arguments and facts, it was urged that mo-
tions and forces, which certainly exist in cases of combustion, would
produce phsenomena exactly similar to those of heat, and therefore
that part of the phsenomena usually attributed to heat are due to
this motion ; and if part of them, probably the whole. And further,
that if the phaenomena of radiation of heat are explained by this
motion of the particles of matter, light is simply radiated heat of
considerable intensity ; and that imponderable substances, whether
under the names of aether, caloric, or phlogiston, are equally ima-
ginary.
Also, a paper was read " On the Question — What is the Solution
of a Differential Equation ?" By Professor De Pvlorgan.
This paper is a short supplement to § 3 of a paper on some points
of the integral calculus (Camb. Trans, vol. ix. part 2). It discusses
the principles on which such an equation as y''^=:a'^, giving
(y —ax + b)(y -\- ax -{- c)=0,
is generally affirmed to be completely solved when 6=c. It dwells
on the distinction between a relation and an equation, which may
express the alternative of one or more relations ; it points out several
cases in which conclusions applicable to the simple relation only are
affirmed of any equation ; and, with reference to the question asked
in the title, discusses the manner in which the answer depends on
the cross-question, what degree of discontinuity is allowed to be im-
plied in the word solution ?
173
May 12, 1856.
A paper was read by Mr. Warburton " On Self- repeating Series,"
in continuation of a former paper.
The author showed in his former paper on self-repeating series,
printed in vol. ix. part 4. of the ' Transactions ' of the Society, that
in the fraction which generates a series of either of the following
forms,
l2n^22".f + 32«.f3+&C. . . ,
or
the numerator of such fraction is a recurrent function of t. He also
then determined the coefficients of the several powers of t in such
numerator to be given linear functions of the differences (as the case
maybe) of 0^', or of 0"'*+^
In his present paper, from the n pairs of equal coefficients which the
recurrent numerator contains, the author obtains n linear equations
between the 2« differences concerned ; and selecting any n of these
differences, he concludes that each of them can be expressed in terms
of the other n differences not so selected ; and consequently that no
formula, expressed in terms of the differences of 0^'* or 0^"■*■^ need
contain more than n of those differences.
He gives the equations requisite for obtaining A'*''"^(0^") in terms
of (A", A'*-^ . . . A2, A 1)0^ ; and a""^'+^(0^'*+'), in terms of
(A"^\ A", . . . A3, A2)0^"'^' ; and he applies these and other of his
equations to the elimination of particular differences of zero from
sundry formulas.
Also, Mr. Bashforth exhibited models illustrating the Moon's
motion.
Also, a paper was read by Mr. Maxwell " On the Elementary
Theory of Optical Instruments."
The object of this communication was to show how the magnitude
and position of the image of any object seen through an optical in-
strument could be ascertained without knowing the construction of
the instrument, by means of data derived from two experiments on
the instrument. Optical questions are generally treated of with
respect to the pencils of rays which pass through the instrument.
A pencil is a collection of rays which have passed through one point,
and may again do so, by some optical contrivance. Now if we sup-
pose all' the points of a plane luminous, each will give out a pencil
of rays, and that collection of pencils which passes through the in-
strument may be treated as a beam of light. In a pencil only one
ray passes through any point of space, unless that point be the focus,
in a beam, an infinite number of rays, corresponding each to some
point in the luminous plane, passes through any point ; and we may.
174
if we choose, treat this collection of rays as a pencil proceeding from
that point. Hence the same beam of light may be decomposed into
pencils in an infinite variety of ways ; and yet, since we regardjit as
the same collection of rays, we may study its properties as a beam
independently of the particular way in which we conceive it analysed
into pencils.
Now in any instrument the incident and emergent beams are com-
posed of the same light, and therefore every ray in the incident beam
has a corresponding ray in the emergent beam. We do not know
their path within the instrument, but before incidence and after
emergence they are straight lines, and therefore any two points serve
to determine the direction of each.
Let us suppose the instrument such that it forms an accurate
image of a plane object in a given position. Then every ray which
passes through a given point of the object before incidence passes
through the corresponding point of the image after emergence, and
this determines one point of the emergent ray. If at any other
distance from the instrument a plane object has an accurate image,
then there will be two other corresponding points given in the inci-
dent and emergent rays. Hence if we know the points in which an
incident ray meets the planes of the two objects, we may find the
incident ray by joining the points of the two images corresponding
to them.
It was then shown, that if the image of a plane object be distinct,
fiat, and similar to the object for two different distances of the object,
the image of any other plane object perpendicular to the axis will be
distinct, flat, and similar to the object.
When the object is at an infinite distance, the plane of its image
is the principal focal plane, and the point where it cuts the axis is
the principal focus. The line joining any point in the object to the
corresponding point of the image cuts the axis at a fixed point called
the /oca/ centre. The distance of the principal focus from the focal
centre is called the principal focal length, or simply th.Q focal length.
There are two principal foci, &c. formed by incident parallel rays
passing in opposite directions through the instrument. If we sup-
pose light always to pass in the same direction through the instru-
ment, then the focus of incident rays when the emergent rays are
-parallel is the first principal focus, and the focus of emergent rays
when the incident rays are parallel is the second principal focus.
Corresponding to these we have first and second focal centres and
focal lengths.
Now let Qi be the focus of incident rays, Pj the foot of the per-
pendicular from Qj on the axis, Q2 the focus of emergent rays, P^ the
foot of the corresponding perpendicular, FjFc^ the first and second
principal foci, AjAg the first and second focal centres, then
PiF^_P,Q>^F,P,
A,Fi P2Q2 F3A3'
lines being positive when uieasured in the direction of the light.
175
Therefore the position and magnitude of the image of any object is
found by a simple proportion.
In one important class of instruments there are no principal foci
or focal centres. A telescope in which parallel rays emerge parallel
is an instance. In such instruments, if m be the angular magnifying
power, the linear dimensions of the image are — of the object, and
m
the distance of the image of the object from the image of the object-
glass is — of the distance of the object from the object-glass. Rules
were then laid down for the composition of instruments, and sug-
gestions for the adaptation of this method to second approximations,
and the method itself was considered with reference to the labours
of Cotes, Smith, Euler, Lagrange, and Gauss on the same subject.
November 6, 1856.
A paper was read by Dr. Donaldson " On the Structure of the
Athenian Trireme, considered with reference to certain difficulties
of Interpretation."
The author's intention was to show in this paper that the Eirrange-
ments for seating the three tiers of rowers in the trireme, which Dr.
Arnold has called " an indiscoverable problem," may be adequately
explained by an examination of the terms which are used to discri-
minate the rowers, and of other words referring to the different parts
of the war-galley. The name of the zygitce, or rowers of the middle
tier, implies that they sat on the ^vya, or transverse planks connect-
ing the opposite sides of the vessel, also called aeXfjtara, and in
earlier times kXtjI^cs. The thalamita, or rowers of the lowest tier,
must, in accordance with their name, have had their seats attached
to the ribs of the vessel in the ddXafxos, or hold. And the thranita,
or rowers of the highest tier, sat on dprjwes, or benches like low
stools, extending for seven feet along the alternate ^vya. The epi-
batee, or marines, whether as working the supernumerary oars, or
as fighting, occupied platforms running along the bulwarks. This
view of the matter explained the fact that there was a gangway
from the stem to the prow for the passage of the officers, &c. along
the ffeX^ara or i^vyd, between the ends of the stools on which the
thranitae sat. This gangway was called the aeXls, and the same
name was given to the passages leading down to the orchestra from
the upper part of the theatre between the rows of seats occupied by
the spectators. Hence was derived a philological explanation of the
words in Aristoph. Equites, 546 : —
oipeaH' avr^ ttoXv to podior, TrapaTreyui^ar' €<}>' ey^eKa Ku)irais
d6pvj3oy yp-qfJToy Xrjvatrrjv'
for there were eleven tiers of seats betweeii each diazoma of the
theatre, which were divided again by the selis ; so that the spectators
176
would represent eleven banks of oars, seated, as in the trireme, with
the lower rows in advance. In the same way, the use of the selis in
a trireme, as the gangway for the officers, &c., explains the lines in
the Agamemnon of ^schylus, 1588-9 : —
av ravTU (j)wv€2s veprepq. Ttrpoarjfxevos
KwTTj/, icparovvTwv Twy eiri ^vy(p hopos ;
for if the zygitce had been intended, they must have been described
as rwv eTTt ^vyiSv. The same view of the GeX^ara, as the proper
place for the officers, was used to explain another passage in the
same play (v. 1413), where Agamemnon's companion is described as
vavTiKQv creXfj-arwy i(rrorpil3ris. And the risk of passing along these
planks, with intervals between them, was considered to explain the
proverbial warning that we must take care not to miss our footing
and fall into the hold (Eurip. Heracl. 168). Other points were in-
cidentally noticed.
November 10, 1856.
The Master of Trinity read a paper " On the Platonic Theory of
Ideas."
In this, he first stated the Platonic theory of ideas as given by the
late Professor Butler of Dublin, in his 'Lectures ' (vol. ii. p. 1 17); he
then remarked that this theory had evidently, for one of its objects,
to explain the possibility of necessary, and therefore eternal truths ;
and thus was an attempt to solve the. problem, often debated in
modern times, of the grounds of mathematical truth ; an attempt
especially called out by the Heraclitean skepticism of Plato's time.
The doctrine of ideas which belong to the intelligible, not to the
visible world, and which are the basis of demonstration, did really
answer its purpose, and account for the existence of real and eternal
truths ; and at the same time, by the tenet that sensible things par-
ticipate in those ideas, accounted for the securing of truth respecting
the sensible world. But when Plato goes on to speak of ideas of
tables and chairs and the like, he gives an extension of the theory
which solves no difficulty, and for which no valid reason is rendered.
The arguments against this extension of the theory are given with
great force in the Dialogue entitled Parmenides, and are not answered
there, nor in a satisfactory way, in any part of Plato's writings.
Moreover, throughout this Dialogue, Parmenides is represented as
having, in his conversation with Socrates, vastly the superiority, not
only in argument, but in temper and manner ; and Socrates and his
friends, after a little show of resistance, assent submissively to all
that Parmenides says. On this ground the writer maintains that
the Dialogue is not Plato's, but anti-Platonic, written probably by an
admirer of Parmenides, and tending to represent Socrates and his
disciples as poor philosophers, conceited talkers, and feeble dis-
putants.
177
This view was further confirmed by arguments drawn from the
external circumstances of the Dialogue.
November 24, 1856.
An account was given by Professor Miller, of the restoration of the
Standard of Weight (vide Proceedings of Royal Society, vol. viii.
Nos. '21, 22).
December 8, 1856.
Mr. Humphry read a paper " On the relations of the Vertebrate
Skeleton to the Nervous System."
He pointed out that the central parts, both of the skeleton
and of the nervous system, are composed of segments placed in
front of, or above one another : those of the former being called
" vertebrae," those of the latter "ganglia;" that the vertebrae
correspond with the ganglia, each vertebra having its appropriate
ganglion ; and further, that the processes, or nerves, emanating
from the central ganglionic portion of the nervous system corre-
spond with, and accompany the processes, or bones, appended to
the central portion of the skeleton, so that the bones appended to
any particular vertebra are generally accompanied by the nerves ema-
nating from the ganglion connected with that vertebra. Hence,
where a difficulty is found in referring a bone to its vertebra, assist-
ance may often be derived from a reference to the nerve or nerves
which accompany that bone. Following the guide thus indicated,
Mr. Humphry would refer the upper extremity, not to one — the
occipital — vertebra, according to the plan of Professor Owen, but to
several cervical vertebrae, forasmuch as it derives its nerves from a
considerable tract of the cervical portion of the cord. For the like
reason, the lower extremity may be regarded as appertaining to
several lumbar and sacral vertebrae. The relations of the bones of
the face to their respective cranial vertebrae were pointed out in ac-
cordance with the distribution of the cerebral nerves. It was shown,
that although the size and shape of the skull are proportioned to the
size and shape of the brain, yet that, as a general rule, the thickness
and weight of the skull are in an inverse ratio to the size of the
brain. A comparison of the different nations of mankind proves,
moreover, that the size of the whole skeleton, as well as that of the
skull, is usually proportioned to the size of the brain ; a well-deve-
loped physique being the natural associate of an ample cerebrum.
PROCEEDINGS
CAMBRIDGE PHILOSOPHICAL SOCIETY.
February 9, 1857.
Prof. Challis gave an account of his Observation of the Occulta-
tion of Jupiter on Jan. 2, 1857.
February 23. 1857.
A paper was read " On the Theory of Polarized Fasciculi, com-
monly known as Haidinger's Brushes." By the Rev. J. Power,
M.A., Librarian of the University.
In this paper the view taken of the subject is similar to that
which had occurred to M. Jamin, and which will be found in Pog-
gendorfFs Annalen, 1849, p. 145, and in the Comptes Rendus, tome
xxvi. p. 197. The author arrived, however, at the present theory
quite independently in the course of last summer, and before he had
acquainted himself with the literature of the subject. M. Jamin
had taken as an essai de calcul the particular semi-visual angle 20°,
which lies far beyond the limits within which the phaenomenon is
visible ; and he has not attempted to give the general law for small-
angles, which was the real problem to be solved.
This is what the author has attempted in the present communica-
tion, availing himself of the experimental researches of Chossat
given in the Bulletin de la Soc. Philomatique, 1818, p. 94.
The subject was rendered more complicated by the circumstance
that the formulae for the intensities of the refracted pencils are given
diflferently by Neumann, Airy, and the author of this paper. Instead
of taking any one set of formulae, the author managed to take them
all into consideration by previously showing that Airy's formulae
result from Neumann's by multiplying them by ^, which is equi-
1 fl tanfl
valent to - . ; while his own result from the same by multi- ;
/i cos Q,
No. XIII. — Proceedings of the Cambridge Phil. Soc.
180
i)lyinK them by °°~ - ; and 0, being the anglei of incidence and
' "^ cotiO,
rcfmrtioJi.
It foIlowH from thenco that, whichever set of formuleo wo may
prefi-r. we Mhall have after refraction, in all casea, the following sim-
j)Ie rchitJDn for two opijositely polarized incident pencils of equal in-
tensity ; namely,
intensity of poncil polmi/id in the piano of incidence
intensity of the pencil polmizcd at right angles
=cos« (d-O,).
Thifl frnrlion drcn .» . - us llic dcviMlioii — 0^ incrcaHCs ; it is
tlicnloic less lor tin; violcl. ra\> llian llic icd. Im the indigo than
for llic \( !lti\v. ;inil iliis :,( ivcs to I'xpliiiu \n ii gciicnd way tljc dingy
yellow . 1 1 i|i< Ml I Ik jihitu ol poliiri/ution, iuul the bright violut stripe
in the i)laue :il i n' lii .ui;' Ic
'I'lif antlioi has .lid . ,.11 iilni (1 the ell'ert in a jilane making on
angle «/> with the pliuu ol jn ali st polarization, and arrives at the
following re.sull : that, |iio\i(l(il we allrihnte to the diHtribution of
the optic nerve Mieli ii vaiialmii nl Hcnsibility as, taken in conjnnc-
tion with the iirlioii of tin- ill , shiill |Md(luee a fiild of vicw nni-
foriiily iuii'hl Iroiii llic cciilic (Hilwanl when c.iiuiiioii daylight is
viewed (a condition wliu^li the author lulieM i (oiiiiiion to all eyct>
with bin own), we Khali have for the brightnc ai any i)oint of the
field of vicw the following • \i>n ssion,
M.(l-ffy^coi2^),
where M is the central brightness, e the degree of polarization (being
for common daylight, and 1 for completely polarized light), and
y= '07309 for rays of mean refrungibility.
The last expression gives us without difficulty the form of the
curves of equal brightness.
Assuming this constant brightness to bo cM, and putting
«a6cos0,
y=Osin0,
we find for the equation sought
The curves are therefore equilateral hyperbolaa havinff tlu> linoH in
octants for their common asymptotes, which confbund themselves
with the curves themselves when ce=0, the case of moan brightness.
The yellow fasciculi have their vcrtici s in the plane of pt)lariza-
tion, and the violet fasciculi have their \( Mice m the plane at right
anffles.
It will be seen that for a given value of B, the bright luss. fur rays
181
of all degrees of refrangibility, that is for all values of y, is least in
the plane of greatest polarization and greatest in the plane at right
angles — contrary to the idea of Moigno, who, for insufficient rea-
sons, imagined that the maximum occurred in the plane of polariza-
tion and the minimum in the plane at right angles.
The yellow tint in the position of minimum intensity, and the vio-
let tint in the position of maximum intensity, is nevertheless per-
fectly accounted for by the consideration that y is greater for the
violet and indigo rays than for the red and yellow.
I'he paper further contains some observations respecting a subjec-
tive centre of the eye, distinct from the usual objective centre, which
may be read with interest, as they remove some difficulties con-
nected with the theory of vision, which had often occurred to the
author, and may have occurred to others.
March 9, 1857.
Mr. Hopkins gave an account of some experiments on the conduc-
tivity of various substances, and pointed out the bearing of the results
on theoriffi of terrestrial heat.
April 27, 1857.
Mr. Humphry read a paper " On the Proportions of the Human
Frame."
May 11, 1857.
A paper was read by Professor Stokes, " On the Discontinuity of
Arbitrary Constants which appear in Divergent Developments."
In a paper " On the Numerical Calculation of a class of Definite
Integrals and Infinite Series " printed in the ninth volume of the
'Cambridge Philosophical Transactions,' the author succeeded in put-
ting the integral I cos - (w^ —mw)dw under a form which admits of
receiving every numerical calculation when m is large, whether po-
sitive or negative. The integral is obtained in the first instance under
the form of circular functions for m positive, or an exponential for
m negative, multiplied by series according to descending powers of
m. ITiese series, which are at first convergent, though ultimately
divergent, have arbitrary constants as coefficients, the determination
of which is all that remains to complete the process. From the
nature of the series, which are applicable only when m is large, or
when it is an imaginary quantity with a large modulus, the passage
from a large positive to a large negative value of m cannot be made
182
through zero, but only by making m imaginary and altering its am-
plitude by TT. The author succeeded in determining directly the
arbitrary constants for m positive, but not for m negative. It was
found that if, in the analytical expression applicable in the case of m
positive, — m were w^ritten for m, the result would become correct
on throwing away the part involving an exponential with a positive
index. There was nothing however to show d, priori that this pro-
cess was legitimate, nor, if it were, at what value of the amplitude of
m a change in the analytical expression ought to be made, although
the occurrence of radicals in the descending and ultimately divergent
series, which did not occur in ascending convergent series by which
the function might always be expressed, showed that some change
analogous to the change of sign of a radical ought to be made in pass-
ing through some values of the amplitude of the variable m. The
method which the author applied to this function is of very general
application, but is subject throughout to the same difficulty.
In the present paper the author has resumed the subject, and has
pointed out the character by which the liability to discontinuity in
the arbitrary constants may be ascertained, which consists in this,
that the terms of an associated divergent series come to be regularly
positive. It is thus found that, notwithstanding the discontinuity,
the complete integrals, by means of divergent series, of the differen-
tial equations which the functions treated of satisfy, are expressed
in such a manner as to involve only as many unknown constants as
correspond to the degree of the equation.
Divergent series are usually divided into two classes, according as
the terms are regularly positive, or alternately positive and negative.
But according to the view here taken, series of the former kind ap-
pear as singularities of the general case of divergent series proceeding
according to powers of an imaginary variable, as indeterminate forms
in passing through which a discontinuity of analytical expression
takes place, analogous to a change of sign of a radical.
A communication was likewise made by the Rev. W. T. Kingsley,
" On the application of Photography to Wood Engraving."
May 25, 1857.
Mr. Bashforth made a communication " On some Calculations and
Experiments undertaken for the purpose of testing the Theories of
Capillary Action."
Also Mr. Candy exhibited a Physiological Alphabet.
The principle of this alphabet is to make the form of the letter
indicate the manner in which the sound is produced, by showing the
position of the organs of speech concerned.
CATTDTS P]Enrsi©I^©GI -\-bc4/ + bs.
If OQP returns to its initial position without performing a com-
plete revolution about O, the limits oi (p and \p are 0, and the area of
the figure traced out by P is bs.
If OQP has performed a complete revolution, the limits of (p and xp
are 27r, and the area traced out is
7r(a^ + 2bc)-\-bs.
A paper was also read by the Astronomer Royal, " On the sub-
stitution of Methods founded on Ordinary Geometry for Methods
based on the General Doctrine of Proportions, in the treatment of
some Geometrical Problems."
The doctrine of proportions laid down in the fifth book of Euclid
is the only one applicable to all cases without exception, but it is
cumbrous and difficult to remember. It is therefore natural to
attempt, in special applications of the doctrine, to introduce the
facilities which are special to each case. This has been done long
193
since in the case of numbers, and this the author of this paper
attempts in some cases in which geometrical lines only are the
subject of consideration, by a new treatment of a theorem equivalent
to Euclid's simple ex ceqiiali and of the doctrine of similar triangles,
referring to nothing more advanced than Euclid, Book II.
The author proves, —
1. If the rectangle contained under the sides a, B be equal to
the rectangle contained under the sides b, A ; and if these rectangles
be so applied together that the sides a and b shall be in a straight
line and that the side B shall meet the side A, the two rectangles
will be the complements of the rectangles on the diameter of a
rectangle.
2. If the rectangle contained under the lines a, B is equal to the
rectangle contained under the lines b, A ; and if the rectangle under
the lines b, C is equal to the rectangle contained under the lines
c, B ; then will the rectangle contained under the lines a, C be
equal to the rectangle contained under the lines c, A.
(This is equivalent to the ordinary ex eequali theorem.
If a:J::A:B
and J : c : : B : C,
then will a:c::A:C.)
3. If two right-angled triangles are equiangular, and if a, A are
their hypothenuses, and b, B homonymous sides, the rectangle con-
tained under the lines a, B is equal to the rectangle contained under
the lines b, A.
(The equivalent theorem in proportions is
a:6::A:B.)
4. If a, c and A, C are homonymous sides of equiangular triangles,
the rectangle contained under a, C will be equal to the rectangle
contained under c, A.
5. U b, c and B, C are homonymous sides including the right
angles of two equiangular right-angled triangles, the rectangle con-
tained under b, C will be equal to the rectangle contained under
c, B.
6. If the rectangle contained under the lines a, B is equal to the
rectangle contained under the lines b, A ; the parallelogram con-
tained under the lines a, B will be equal to the equiangular paral-
lelogram contained under the lines b, A.
(This is equivalent to the proposition.
If a : 6 : : A : B
then a:b:: A cos a : B cos a.)
These propositions will suffice for the treatment of the first
thirteen propositions of Euclid's sixth book (Prop. I. excepted), and
of all the theorems and problems apparently involving proportions
of straight lines (not of areas, &c.) which usually present themselves.
The author then proceeds, as an instance of their application, to
prove by means of them the following theorem : —
194
If pairs of tangents are drawn externally to each couple of three
unequal circles, the three intersections of the tangents of each pair
will be in one straight line.
Also a paper was read by Professor De Morgan, " On a Proof
of the existence of a Root in every Algebraic Equation : with an
examination and extension of Cauchy's Theorem of Imaginary
Roots ; and remarks on the proofs of the existence of Roots given
by Argand and by Mourey."
The extension of Cauchy's theorem is very easily found, when the
proof is the first of those given by Sturm in Liouville's Journal.
The extended theorem is as follows : —
Let (pz be any function of z, and let 2=jr-|-y a/ — 1. Let (x, y)
be a point on any circuit which does not cut itself. Let this point
describe the circuit in the positive direction of revolution ; and,
p
osition
of a point very near the sun's limb ; and in fact M. Lescarbault does
not appear to have attempted to determine the position of the spot
during the interval between the beginning and the end of the
transit. He states that the spot had entered a little way on the sun
when he first saw it, and that the time and place of entrance were
inferred by estimation.
The following are the immediate results of the observations : —
The spot entered at 4^ 5"" 36* mean time of Orgeres at the angular
distance of 57° 22' from the north point towards the west, and de-
parted at 5*^ 22'" 44*, at 85° 45' from the south point towards the
west, occupying consequently in its transit l*" 17*" 8\ The length
of the chord it described was 9' 14", and its least distance from the
sun's centre 15' 22". M. Lescarbault also states that he judged the
apparent diameter of the spot to be at most one-fourth of that of
Mercury, when seen by him with the same telescope and magnifying
power during its transit across the sun on May 8, 1845. The lati-
tude of Orgies is 48° 8' 55", and longitude west of Paris, 2™ 35*.
From these data M. Leverrier ascertained, by calculating on the
hypothesis of a circular orbit, that the longitude of the ascending
node is 12° 59', the inclination 12° 10', the mean distance 0-1427,
that of the earth being unity, and the periodic time 19*7 days. Also
he found that the greatest elongation of the body from the sun is 8°,
the inclination of its orbit to that of Mercury 7°, the real ratio of
its diameter to Mercury's 1 to 2*58, and that its volume is one-
seventeenth the volume of Mercury on the supposition of equal den-
sities. This mass is much too small to account for the perturbation
of Mercury's perihelion. According to these results, the periods at
which transits may be expected are eight days before and after
April 2 and October 5, the body being between the earth and sun
221
near its descending node at the former period, and near its ascend-
ing node at the latter.
After the announcement of this singular discovery, it was found
that other observations of a like kind had been previously made.
Several instances are collected by Professor Wolf in the tenth num-
ber of his Mittheilungen ilber die Sonnenjlecken, eight of which are
quoted in vol. xx. (p. 100) of the Monthly Notices of the Royal
Astronomical Society, Two of these, the observation of Stark on
October 9, 1819, and that of Jenitsch on October 10, 1802, agree
sufficiently well with the calculated position of the node of the object
seen by Lescarbault. But the spot seen by Stark is stated to have
been about the size of Mercury,
Capel LofFt saw at Ipswich, on January 6, 1818, at 11 a.m., a
spot of a ' sub-elliptic form,' which advanced rapidly on the sun's
disk, and was not visible in the evening of the same day (Monthly
Magazine, 1818, part 1, p. 102).
Mr. Benjamin Scott, Chamberlain of London, saw about mid-
summer of 1847 a large and well-defined round spot, comparable in
apparent size with Venus, which had departed at sunrise of the next
day (Evening Mail, January 11, 18G0).
PastorfF of Buchholz records that he saw on October 28 and
November 1, 1836. and on February 17, 1837, two round black
spots of unequal size, moving across the sun at the respective hourly
rates of 14', 7", and 28'. Also he announced, January 9, 1835, to
the Editor of the Astronomische Nachrichten, that "six times in the
previous year he had seen two new bodies pass before the sun in dif-
ferent directions and with different velocities. The larger was about
3" in diameter, and the smaller from 1" to 1"'25. Both appeared
perfectly round. Sometimes the smaller preceded, and at other
times the larger. The greatest observed interval between them was
1' 16": at times they were very near each other. Their passage
occupied a few hours. Both appeared as black as Mercury on the
sun, and had a sharp round form, which, however, especially in the
smaller, was difficult to distinguish." Schumacher considered it his
duty as editor to insert the communication, but evidently did not
give credit to it (Astron. Nachr. No. 273).
In vol. ii. of the Correspondence between Olbers and Bessel,
mention is made in p. 162 of an observation at Vienna by Steinhiibel,
of a dark and well-defined spot of circular form which passed over
the sun's diameter in five hours. Olbers, from these data, estimates
the distance from the sun to be 0*19, and the periodic time thirty
days. It is remarkable that Stark saw about noon of the same day
a singular and well-defined circular spot, which was not visible in
the evening. Tliis is one of the instances in vol. xx. of the Monthly
Notices of the Astronomical Society.
These accounts appear to prove that transits of dark round objects
across the sun are real phaenomena ; but it would perhaps be prema-
ture to conclude that they are planetary bodies. If the object ob-
served by Lescarbault be a planet, it is certainly very surprising that
it has not been often seen. Schwabe, after observations of the sun's
222
face continued through thirty-three years, has recorded no instance
of such a transit. It is probable that now attention has been espe-
cially drawn to the subject, future observations, accompanied by
measures (of which Lescarbault's are the first instance), may throw
light on the nature of these phsenomena.
April 2-4, 1860.
Professor De Morgan read a paper " On the Syllogism, No. IV.,
and on the Logic of Relations."
In the third paper were presented the elements of a system in
which only onymatic relations were considered ; that is, relations
which arise out of the mere notion of nomenclature — relations of
name to name, as names. The present paper considers relation in
general. It would hardly be possible to abstract the part of it
which relates to relation itself, or to the author's controversy with
the logicians, who declare all relations material except those which
are onymatic, to which alone they give the name oi formal. Mr.
De Morgan denies that there is any purely formal proposition except
" there is the probability a that X is in the relation L to Y;" and
he maintains that the notion ' material ' non suscipit magis et minus ;
so that the relating copula is as much materialized when for L we
read identical as when for L we read grandfather.
Let X . . LY signify that X stands in the relation L to Y ; and
X . LY that it does not. Let LM signify the relation compounded
of L and M, so that X . . LMY signifies that X is an L of an M of Y.
In the doctrine of syllogism, it is necessary to take account of
combinations involving a sign of inherent quantity, as follows : —
By X . . LM'Y is signified that X is an L of every M of Y.
By X . . L^MY it is signified that X is an L of none but Ms of Y.
The contrary relation of L, not -L, is signified by /. Thus X . LY is
identical with X . . /Y. The converse of L is signified by L~^ : thus
X . . LY is identical with Y..L~^X. This is denominated the
Ij-verse of X, and may be written LX by those who prefer to avoid
the mathematical symbol.
The attachment of the sign of inherent quantity to the symbols of
relation is the removal of a difficulty which, so long as it lasted, pre-
vented any satisfactory treatment of the syllogism. There is nothing
more in X . . LM'Y than in every M of Y is an L~' of X, or
MY))L~'X, X and Y being individuals; and nothing more in
X . . L.MY than in L~^X))MY, except only the attachment of the
idea of quantity to the combination of the relation.
When X is related to Y and Y to Z, a relation of Xto Z follows:
and the relation of X to Z is compounded of the relations of X to Y
and Y to Z. And this is syllogism. Accordingly every syllogism
has its inference really formed in the first figure, with both premises
aflarmative. For example, Y . LX and Y . . MZ are premises stated
223
in the third figure: they amount to X..L~'Y and Y..MZ,
giving X . . l~^MZ for conclusion. This affirmative form of conclu.
sion may be replaced by either of the negative forms X . L~*M'Z or
X.J,-'mZ.
The arrangement of all the forms of syllogism, the discussion of
points connected with the forms of conclusion, the extension from
individual terms in relation to quantified propositions, the treatment
of the particular cases in which relations are convertible, or transi-
tive, or both — form the bulk of the paper, so far as it is not contro-
versially directed against those who contend for the confinement of
the syllogism to what Mr. De Morgan calls the onymatic form.
An appendix follows the paper, on syllogism of transposed quan-
tity, in which the number of instances included in one premise is
equal to the whole number of existing instances of the concluding
term in the other premise.
Mr. J H. Rohrs also read a paper " On the Motion of Bows, and
thin Elastic Rods." •
May 7. 1860.
The Rev. Professor Sedgwick made a communication " On the
Succession of Organic Forms during long geological periods ; and
on certain Theories which profess to account for the origin of new
species."
May 21, 1860.
The Public Orator read a paper " On the Pronunciation of the
Ancient and Modern Greek Languages."
He gave a rapid sketch of the " Reuchlin and Erasmus" contro-
versy in the sixteenth century, especially the part taken in it at
Cambridge by Cheke, Smith, Ascham, and Bishop Gardiner ; and
then proceeded to show how the proper sounds of the Greek letters
may be determined from the following sources : —
1. Distinct statements of grammarians.
2. Incidental notices in other ancient authors.
3. Variations in writing of inscriptions and MSS.
4. Phonetic spelling of cries of animals.
5. Puns and riddles.
6. The value of the respective letters in other languages employ-
ing the same alphabet, especially Latin.
7. The way in which Latin proper names are spelt in Greek, and
vice versa.
8. The traditions of pronunciation preserved in modem Greek.
He concluded that, on the whole, the method of Erasmus ap-
proached more nearly to the ancient pronunciation than that of
Reuchlin.
224
" But," he proceeded, " when we consider the untrustworthiness
of each of these sources of evidence taken singly, and when moreover
we find them often in conflict with one another, it cannot be ex-
pected that the result should be very certain or very satisfactory.
There are also other considerations which enhance the difficulty of
the inquiry. As there were very marked dialectic varieties in Greece,
so there may have been local variations even in Attica itself.
" The pronunciation, too, changed from time to time. Plato gives
us proof of this in the ' Cratylus.' "
After quoting several instances, and showing that great changes
both in pronunciation and spelling had taken place in modern lan-
guages, French, Spanish, and English, " it would," he said, " be
hopeless to attempt to determine the pronuilfciation of any language
by a reference to its orthography at a time when both were perpe-
tually changing. But in the history of every nation there arrives a
time when the creative energy of its literature seems to have spent
itself; when, instead of developing new forms, men begin to look
back and not forward, to comment and to criticise. Then it is that
a language begins to assume, even in minor and merely outward
points, such as pronunciation and spelling, a fixity and rigidity
which it retains with scarcely any change so long as the nation
holds together. Such a period in Greek history was that which
began with the grammarian sophists in the fifth century B.C., and
culminated in Aristarchus and Aristophanes of Byzantium. In the
spelling and pronunciation of Greek there was probably very little
change from that time to the end of the third century a.d."
October 19. 1860.
Dr. Paget made a communication " On some Points in the Physio-
logy of Laughter."
November 12, 1860.
The Public Orator read a paper (a sequel to that on May 21) "On
the Accentuation of Ancient Greek."
The question of accents was not discussed in the Reuchlin and
Erasmus dispute. At that time all pronounced according to the
system of accents introduced by the Greeks of Constantinople, who
first taught the ancient language to the Italians.
It was probably in Elizabeth's reign that we began to disuse the
old pronunciation of vowels both in Greek and Latin ; and concur-
rently with this change we, as well as the other nations of Europe,
began to pronounce Greek, not with the modern Greek, but with
the Latin accent. The reasons were : —
1 . Teachers speaking the modern Greek were no longer required,
so the tradition was not kept up.
225
2. It saved much trouble to pronounce both languages with the
same accentuation.
3. The Greek accent perpetually clashes with quantity; the
Latin much more rarely ; never, indeed, in that syllable of which
the quantity is most marked — the penultima.
Isaac Vossius (1650-60) advocated the disuse of accentual marks
altogether, as the invention of a barbarous age to perpetuate a bar-
barous pronunciation.
After showing the meaning of the word * accent ' as applied to
modern languages, and discussing the accentuation of the German,
English, French, &c., he proceeded to say :
"There are three methods of emphasizing a syllable: —
1. By raising the note ;
2. By prolonging the sound ;
3. By increasing its volume.
" Scaliger, De Causis Linguce LatincB, lib. ii. cap. 52, recognizes
this division when he says that a syllable may be considered of three
dimensions in sound, having height, length, and breadth.
" Now in our own language, when we accent a syllable, which
of these dimensions do we increase ? Generally all three, but not
necessarily ; for when the prayers, for example, are intoned, i. e.
read upon one note, the accent is marked by increasing the volume
of sound (the third method), which involves also a longer time in
utterance, i. e. a lengthening of quantity. In speaking, all three
methods are employed, but one more prominently than the other,
according to individual peculiarities of the speakers. What we
blend, the Greeks kept distinct.
" We cannot understand the Greek system unless we bear this in
mind. They never confounded accent with quantity. Ineradicable
habit prevents us from reverting in practice to their method, just as
they would have been unable to comprehend ours.
" It is clear from Dionysius, De Comp. Verb, lib, xi. cap. 75, that
the dialogue in tragedy presented the ordinary accentuation, which
was disregarded only in choral passages set to music."
The practical conclusion was this : that while it would be desirable,
if possible, to return to the Erasmian system of pronunciation, it
would be extremely absurd to adopt the barbarous accentuation of
modern Greek, which has quite lost the old essential distinction
between accent and quantity. In this respect, as we cannot recover
practically the ancient method, it is better to keep to our own system
of the Latin accent, which does not confuse the learner's notion of
quantity in verse as the modern Greek does.
An Athenian boy has the greatest difficulty in comprehending the
rhythm of Homer or Sophocles. Hence it is not blind prejudice
(as Professor Blackie asserts) which makes us keep to our old usage,
but a well-grounded conviction that we should lose more by changing
than we should gain.
226
November 26, 1860.
Professor ChaUismade a communication " On the Solar Eclipse of
July 18, 1860."
December 10, 1860.
Mr. Seeley read a " Notice of Opinions on the Red Limestone
at Hunstanton."
Professor Miller also described " An Instrument for measuring the
radii of arcs of Rainbows."
February 11, 1861.
Mr. H. D. Macleod read a paper " On the present State of the
Science of Political Economy."
The writer took a general survey of the science as it at present
exists, testing several generally received doctrines by the principles
of inductive logic, and earnestly enforcing the necessity of a thorough
reform of the whole science, which must be constructed on prin-
ciples analogous to those of the other inductive sciences.
February 25, 1861.
Dr. Humphry made a communication " On the Growth of Bones.
March 11, 1861.
The Master of Trinity made a communication " On the Timaeus
of Plato."
PROCEEDINGS
OF THE
CAMBRIDGE PHILOSOPHICAL SOCIETY.
October 26, 1863.
A communication was made by Mr. H. D. Macleod *' On the
Theory of Banking."
November 9, 1863.
Communications were made by Dr. Humphry, " The Results of
Experiments on the Growth of the Jaw."
By Mr. Todhunter " On a Question in the Theory of Probabilities."
November 23, 1863.
A communication was made by Professor Challis " On the Meteor
of August 10, 1863."
December 7, 1863.
A communication was made by Dr. Akin " On the Origin of Elec-
tricity."
February 8, 1864.
A communication was made by Professor Liveing " On the new
Metal, Thallium."
No. XVI. — Proceedings of the Cambridge Phil. Soc.
228
February 22, 1864.
Communications were made by Professor Cayley " On a Case of
the Involution of Cubic Curves," and " On the Theory of Involution."
By Mr. Spencer " On Vitality."
March 7, 1864.
Communications were made by Mr. Harry Seeley —
1. " On the Significance of the Succession of Rocks and Fossils."
2. " On Pterodactyles, and on a new Species of Pterodactylus
mach(Brorhynchus."
April 18, 1864.
Communications were made by Mr. Rohrs " On the Strains and
Vibrations of Ordnance under the Action of Gunpowder."
By Professor Cayley " On the Classification of Cubic Curves,"
and "On Cubic Cones and Curves."
May 2, 1864.
Communications were made by Mr. Harry Seeley —
1. "Note on Palceohalcena Sedgwicki (nob.), a Whale from the
OoUte."
2. " On the Fossil Birds of the Upper Greensand, Palceocolyntus
Barretti (nob.) and Pelargonis Sedgwicki (nob.)."
3. " On the Osteology and Classification of Pterodactyles, Part II.,
with descriptions of the new species P. Hopkinsi and P. Oweni."
May 16, 1864.
Papers were read by Professor De Morgan —
1. "A Theorem relating to Neutral Series."
The theorem is as follows. If a^—a^-\-a^— ... be a convergent
series which has the limiting form 1 — 1 + 1 — ... , and if a« be of
229
continuous law, so that er^+i : «« finally approaches towards a limit,
the limit towards which the series approaches as its form approaches
neutrality is Oz—Uz+i divided by 02—02+2. And this limit is
always |-. If there be in the series a cycle of laws involving an even
number of terms, so that
02n« — (^2nz+l> ^2nz+\ — 02ns+2> • • • 02nz+2n — l — 02nz+in
approach in ratio to Aq, A;^, ...Z:2n-i. then the two series 0^ — fli + Oj — •••
and a^ — a.^ + Og— . . . , which have unity for their sum, have the
ratio of k^-\-k^+ . . . +k2n-2 to k^-^k^-\- . . . -^-kin-i- But if the
cycle have an odd number of terms, each of these series is g, just as
if the law had been continuous. The demonstration is founded upon
the following theorem : — If Po + Pj+ . . . and Qq + Qj+ • • • be di-
verging series, whether of increasing or decreasing terms, their two
infinite sums are in the final ratio of Pz to Q^. Applications of this
theorem are given to the determination of a large number of terms
of l"-|-2"+ ... when w is — 1 or greater, and to the determination
of the usual approximation to 1 . 2 . 3 . . . « when n is great.
2. " On Infinity, and on the Sign of Equality."
The author professes himself satisfied of the subjective reality of
the notions of infinitely great and infinitely small. His paper, as far
as it deals with various objections by various modes of answer, is
not capable of abstract ; but four points, on which he especially
relies, may be stated as follows : —
1. The concepts of the mind are divisible into imageable and un-
imageable : the first can be pictured and placed before the mind's
eye ; the second cannot. The mathematician, dealing in great part
with imaged concepts, is apt to repel the unimageable, as if it
could not be a legitimate object of mathematical reasoning. But all
that is necessary to reasoning is knowledge of the connexion of sub-
jects and predicates. Infinite quantity is unimageable in its relation
to finite quantity, but not therefore inconceivable, nor destitute of
known attributes. A million of cubic miles is as destitute of image
as infinite space ; nevertheless it is a conception the attributes of
which give known propositions.
2. Number, or enumeration as distinguished from multitude, is a
concept from which no notion of infinity can be gained ; but much
perplexity has arisen from the attempt to make it a teacher of this
subject. Abstract number has more than one aflFection which is de-
rived from the concrete in such manner that the two abstractions,
number and its aifection, cannot have their function explained ex-
cepting by return to the concrete. Such aflPections are the divisible
unit, on which the doctrine of fractions is founded, and the opposi-
tion of positive and negative. The representation of infinite and of
infinitesimal number is a third . affection of the numerical, which
cannot be explained on purely numerical notions.
3. The infinite is not a kind of terminus to the finite, but another
status of magnitude, such that no finite, however great, is anything
but an infinitesimal of the infinite. And the same may be said of
each order of infinity with reference to the one below it.
230
4. The symbol ^, the infinite of common algebra, represents an
extreme of infinite which can no more be attained by passage through
orders of infinity, than any infinite by passage from finite to finite.
Each of these positions comes into conflict with some of the usual
arguments for or against the introduction of infinites.
The second part of the paper is on the meaning of the sign of
equality. Mr. De Morgan contends for the ultimate attainment of
a purely formal algebra, in which every transformation shall have
meaning and validity in every possible case. He points out certain
difliculties and inconsistencies in the ordinary use of the sign of
equality, which can, he aflirms, receive a consistent explanation on
the extension which he proposes, and which, to some extent, he con-
siders as virtually adopted.
His notion is that equality, strictly so called, is but a species of the
genns undistinguishable ; and that the actual use of the sign (=)
shows a leaning to the generic definition. Every order of infinites
or infinitesimals has its own metre, and the sign (=) indicates undis-
tinguishability with reference to the metre, which is often in thought,
but for which no symbol is employed. Algebraical changes may or
may not demand or permit changes in the metre. It would be im-
possible to give any further account, with justice to the subject, in
a short abstract.
3. By Mr. Harry Seeley " On Sauromia, and the Classification of
Pterodactyles, Part HI."
October 31, 1864.
Papers were read by Professor Selwyn " On Autographs of the
Sun."
By A. R. Catton, B.A., St. John's College, " On the Constitution
of Chemical Compounds."
November 14, 1864.
A communication was made by G. F. Browne, M.A., St. Catha-
rine's College, " On certain Ice-Caverns."
November 28, 1864.
Papers were read by Dr. Humphry on the question, " Is the Ver-
tebral Theory of the Skull to be abandoned ?"
This communication was intended partly as a reply to the opinion
expressed by Professor Huxley in his lectures on comparative ana-
tomy, that the vertebral hypothesis of the skull has been abolished
by the recent discoveries in development. Dr. Humphry commencd
by calling attention to the Laws of Uniformity of Plan, and Variety
in Detail, which prevail throughout the animal kingdom, and, indeed,
throughout the material system, and which the recent discoveries
231
by the microscope have shown to rule over the ultimate structure
and formation of all the tissues of the body. The discovery of the
illustration of these laws in the plan of cell-formation of the tissues,
and in the development of all animal and vegetable structure from
the simple cell-form, he regarded as the grandest discovery in phy-
sical science that has taken place in our time. Of late years, the
attention of anatomists has been much directed to the exemplifica-
tion of these laws in the vertebrate classes, to tracing the uniformity
of plan, especially in the skeleton, through the variety in detail which
the members of these classes exhibit. This constitutes the branch
of anatomy called " Homology." The general features of the plan
upon which vertebrate animals are constructed are clear enough in
all of them. Osseous segments, or vertebrse, with neural and vis-
ceral processes, enclosing respectively the neural and visceral cen-
tres, constitute the trunk, including neck, chest, loins, &c. Proba-
bility is in favour of the view propounded by Goethe and Oken, and
worked out by Oken and Owen, that the skull falls in with the Law
of Uniformity, and corresponds with the rest of the frame in having
a vertebral composition. It is by all anatomists admitted to be
segmentally constructed. Most anatomists are agreed as to the
number of segments. Ought not, therefore, these segments to be
described by the same name as those of which they form a conti-
nuation, especially as they bear the same relations to the neural
and visceral centres, and the same or nearly similar relations to the
nerves and blood-vessel ? In their mode of development, too, the
segments of the skull show a marked general correspondence with
those of the trunk. The chorda dorsalis, around which the verte-
bral centres are formed, extends at any rate halfway along the base
of the skull ; and the bodies and arches of the cranial segments are
evolved from a continuation of the same embryonic structure (the
" vertebral plates ") as the trunk segments — the chief diflference
being that in the trunk segmentation takes place at an earlier period
than in the head. In the trunk, it is observed in the vertebral plates ;
and these primitive segments are called " protovertebrae." They ap-
pear not to exist in the head. The segmentation, however, takes
place in the cranium as soon as ossification begins, even if it does
not do so before ; and the significance of the protovertebrae as dis-
tinctive features between the skull and the trunk is diminished,
first, by their being related to the formation of the nerves as much
or more than to that of the vertebrae ; and secondly, by their not
really corresponding with the vertebrae, each permanent vertebra
being formed by a half of two protovertebrae. Dr. Humphry expa-
tiated on this and other points in the development of the skull, and
expressed his decided opinion that the differences between the deve-
lopment of it and of the trunk vertebrae were by no means suflftcient
to controvert the view — which coincides with the Law of Uniformity,
and which is confirmed by the segmental construction of the skull,
by the relation of its components to surrounding parts, and by so
many fundamental resemblances in development — that the same
name may be applied to the segments of the skull and of the trunk.
232
and that the one, as well as the other, consists of vertebrae modified
to meet the requirements of the parts in which they are found. He
concluded by stating that the greater number of those anatomists to
whose observations we are indebted for most of our knowledge of
the development of the skull and of the trunk, are agreed that the
differences between the mode of formation of the segments in the
two form no real argument against the vertebral character of either ;
and he thought stronger reasons must be adduced than had yet been
shown before the anatomists could be called upon to abandon the
vertebral theory of the skull.
By Professor De Morgan " On the Early History of the Signs +
and -."
An account is given of the work on arithmetic of John Widman,
printed in 1489, in which the signs + and — are used to denote
more and less. The use made is twofold : a + ft signifying that b
more than a is wanted, infers a direction to add b to a. But a-\-b
in the old rule oi false position is used to signify that the assumption
of a for the answer gives b too much in the solution. This last
usage was continued by many writers through the greater part of the
sixteenth century.
Some account is given of the Die Coss of Chr. Rudolf, which
passes for the first work in which + and — are used. The first
edition of his work being lost, a question is raised as to how far
the second edition, edited by Stifel, is a fair reprint of the first. A
Latin translation of this first edition is said to be in the Imperial
Library at Paris.
From the mannner in which Widman introduces his signs, Mr.
De Morgan thinks there is some ground to suspect that they were
originally warehouse marks, indicating the scale into which smaller
weights were introduced to make the balance, when the nearest
number of larger weights had been put in. This point and others
require the examination of older works, print and manuscript.
February 13, 1865.
Communications were made by Professor Cayley " On Abstract
Geometry."
By Professor Clifton," Note on the Early History of the Signs +
and — ."
February 27, 1865.
Mr. Alfred Newton, M.A., F.L.S., communicated some "Notes
on Spitzbergen," of which the following is an abstract.
The author stated that last summer he accompanied Mr. Edward
Birkbeck on a voyage to Spitzbergen, in that gentleman's yacht, the
* Sultana,' R.S.Y.C. After giving a slight sketch of some of the
principal voyages which had been made to that country, he pro-
233
ceeded to say that the ' Sultana ' left Hammerfest on the 3rd of
July, in company with a Norwegian sloop which had been specially
fitted to encounter the ice, and chartered to attend upon the yacht.
On the 6th they arrived at the entrance of Stor Fjord, which was
found to be entirely blocked by the ice. Horn Sound and Bell
Sound were subsequently discovered to be in the same condition.
They then made for Ice Sound, and anchored in Safe Haven on the
9th. Ice Sound was described as a very much larger inlet than it is
represented in the charts to be, extending at least fifty miles into
the interior. All the valleys on the north side, and consequently
having a southern aspect, are completely occupied by large glaciers,
which, with one exception, are only terminated by the sea. The
single exception consists of a small but remarkable glacier suspended
on a hill-side, some 360 yards from the beach, resting conformably
on its own moraine, and having no apparent means of discharge.
The author supposes this last was effected by filtration through its
bed. The south side of Ice Sound contains several bays of consi-
derable size ; and the valleys opening upon it, and therefore having
a northern aspect, were entirely free from glaciers ; the observa-
tion being directly opposed to the account given of the Spitzbergen
glaciers by Sir John Richardson in his ' Polar Regions.' A great
many reindeer frequent this part of the country, and countless numbers
of sea-fowl breed on such of the high cliiFs around the Sound as are
inaccessible to the Arctic foxes. In Ice Sound Mr. Birkbeck's party
had the pleasure of meeting the Swedish Scientific Expedition under
Professor Nordenskjold, who are engaged in measuring an arc of the
meridian. On the 4th of August the party separated, some going
to the eastward in the Norwegian sloop, while the yacht made an-
other ineff'ectual attempt to ascend the Stor Fjord. The sloop sailed
as far as Ryklis Islands, but was stopped by the ice. She then pro-
ceeded further east in the hope of getting round the pack, and came
in sight of " Commander Gile's land," the existence of which had
been so long doubted, it having been ignored by Sir John Richard-
son in his work, and in the Admiralty Chart of Spitzbergen. It
appears to lie about sixty miles east of the entrance to Walter Thy-
men's Strait, and its flat or round-topped hills (so different from those
of Spitzbergen) were very plainly seen. The author stated that in
1859 the master of a Norwegian vessel landed upon it, and he pro-
duced a pebble which had then been brought thence, in proof of the
reality of its existence as land, and not either ice or fog-bank. He
then proceeded to remark on the driftwood with which the shores of
the " Thousand Islands " are strewn, which he believed to be cer-
tainly of Siberian origin, and not brought, as sometimes imagined,
by the Gulf-stream, — stating that though often worm-eaten, he had
never observed any signs of barnacles upon it. He then commented
on the discovery of the passage from the top of Stor Fjord to Hin-
lopen Strait, of which there had long been a traditionary knowledge,
though it was not effected till 1859 ; and showed, from the Swedish
surveys in 1861, that this passage must lie some thirty miles further
south than the position assigned to it on the Admiralty charts, thus
234
affording another instance of our imperfect knowledge of the geo-
graphy of Spitzbergen. In conclusion, the author stated that he left
Spitzbergen on the 21st of August, the sun having set a night or two
previously for the first time, and the salt water begun to freeze ;
and he warmly urged his audience to support the further circumpolar
exploration which has been lately proposed by Captain Sherard
Osborne ; and said that, as a zoologist, he could declare there were
many questions of the very highest interest which could only be
solved by a new Arctic expedition.
Professor Cardale Babington and Mr. Harry Seeley made commu-
nications respectively on the plants and on the fossils brought by
Mr, Newton from Spitzbergen.
March 13, 1865.
A communication was made by Professor Liveing " On Gun-
cotton."
March 27, 1865.
A communication was made by Professor Miller *' On the Crystal-
lographic Methods of Grassman, Hessel, Frankenheim, and Uhde,
and on their employment in the investigation of the general geo-
metrical properties of Crystals."
May 1, 1865.
Communications were made by Mr. Harry Seeley —
1. "On the Cambridge Greensand. — Part I. The Rock and its
Origin."
2. " On the Gravel and Drift of the Fenland.— Part II. Theory."
May 15, 1865.
A communication was made by Professor Churchill Babington
" On the Coinage of England before the Norman Conquest."
May 29, 1865.
A paper was read by Mr. Todhunter " On the Method of Least
Squares."
The object of this communication is principally to demonstrate a
very remarkable result which Laplace enunciated, without demon-
stration, in the first Supplement to his work on ' Probabilities.' An
exposition is also given of the process adopted by Laplace for inves-
tigating the method of least squares. Laplace's process is genera-
lized and extended ; and results which he obtained for the case of
two elements are shown to hold for any number of elements. The
mathematical part of the investigation consists chiefly in the evolu-
tion of certain definite multiple integrals.
235
October 30, 1865.
The following officers were elected : —
President Rev. W. H. Cookson, D.D.
r Mr. I. Todhunter.
Vice-Presidents. < Dr. Paget.
[^Professor Challis.
Treasurer .... Rev. W. M. Campion.
f Professor Cardale Babington.
Secretaries . . < Professor Liveing.
tRev. T. G. Bonney.
r Professor Sehvyn.
New Members I Rev. W. G. Clark.
the Council, j Mr. R. Potter.
(^Rev. N. M. Ferrers.
The following communications were made to the Society : —
By Mr. A. R. Catton—
1. " On the Synthesis of Formic Acid."
2. " On the possibility of accounting for the double refraction of
Light by the vibrations of a continuous elastic medium kept in a state
of constraint by the action of the material molecules,"
By Professor Cayley —
3. " A new Theorem on the Equilibrium of four Forces acting
on a solid Body."
Defining the " moment of two lines " as the product of the short-
est distance of the two lines into the sine of their inclination, then,
if four forces acting along the lines 1, 2, 3, 4 respectively are in
equilibrium, the lines must, as is known (Mobius), be four genera-
ting lines of an hyperboloid ; and if 12 denote the moment df the lines
1 and 2, and similarly 13 the moment of the lines 1 and 3, &c., the
forces are as
y/23 . 34 . 42 : ^/34.41.13 : v/41 .12.24: -v/l2.23.31.
Calling the four forces V^, P^, P,, P^, it follows as a corollary that
we have
P^P, . 12=12 . 34yi3T42 . x/l4.23=P,P, . 34 ;
viz. the product of any two of the forces into the moment of the
lines along which they act is equal to the product of the other two
forces into the moment of the lines along which they act, — which is
equivalent to Chasles's theorem, that, representing a force by a finite
line of proportional magnitude, then in whatever way a system of
forces is resolved into two forces, the volume of the tetrahedron
formed by joining the extremities of the two representative lines is
constant.
236
November 13, 1865.
By Professor Sedgwick, F.R.S., "A Sketch of the Geology of the
Valley of Dent, with some account of a destructive Avalanche which
fell in the year 1752."
The valley of Dent lies in the north-west corner of Yorkshire,
which is thrust in between Westmoreland and Lancashire, beyond
the natural limits of the county. The upper part of the valley is
excavated in the carboniferous groups which are continued south-
wards into Nottinghamshire, and northwards into Durham and
Northumberland, and through the greatest part of their range form
the watershed between the east and west coasts of England. All
the valleys that drain down to the Lune are partly formed in rocks
of the carboniferous age. In the upper part of Dent Dale, which is
one of these tributaries, the great scar-limestone appears only near
the bottom of the valley, while the sides are formed of soft shale
alternating with harder bands of sandstone and limestone ; and the
whole series is capped by mill-stone grit. The rainfall in some por-
tions of the Lake mountains is not less than 150 or 160 inches in the
year. Among the neighbouring carboniferous mountains the rain-
fall is much less ; but still it is at least three times the English
average ; and the winter fall of snow is in some years enormous.
Hence the becks, or mountain-streams, are often greatly swollen, and
the gills, or lateral branches, frequently descend in brawling torrents
from the mountain- side into the lower valley through deep ravines
and lateral valleys that have been excavated out of the shales and
sandstones in the course of past ages. On rare occasions a great fall of
snow, accompanied by a violent wind, will almost fill up the ravines
and lateral valleys, and form a dam across the descending water ;
and should there be a sudden thaw afterwards, the descending gills
may be held up for a while till the pressure of the water drives down
the barrier, and an avalanche is formed of mingled snow and water
(provincially called a brack), which rushes down with the roar of
thunder, and bears all before it into the beck below. On the 6th
of February, 1752, a very large one fell, destroying several houses
and farm buildings, and killing seven people, besides several head oi
cattle. The following letter, written by an eye-witness, describes
the catastrophe (the spelling and punctuation have been slightly
modernized) : —
" Harbourgill, 6th of the 2nd month, 1752.
"DearBro' and Sister,—
" These few lines I hope will find excuse : for it's not without a
cause that I have written no sooner to you. I fully purposed to
have seen you a considerable time since : but now, as things are at
present, I have lost all hopes of coming. Yet through the good provi-
dence of Heaven we are all alive and pretty well in health : which
is more than could be well expected, considering what dismal times
it has been with us in Dent. I hope I shall never live to see the like
again : for we had the greatest storm of wind and snow that conti-
237
nued for above a week with very little intermission : so that all the
watercourses, both in the mountains and elsewhere, were made level ;
the like never being remembered, for it excited the curiosity of
several persons to view them with wonder and astonishment : yet
little thinking that the consequences would have been so tragical to
many. For at the breaking up of the storm (i. e. frost) it began to
rain exceedingly in the evening, which continued all night and the
next day to that degree that, by 11 o'clock, the dismal scene began.
For the snow in the watercouses being no longer able to sustain the
great quantities of water, all began to slide down the mountains
together with incredible swiftness, driving great rocks, stones, and
earth, all before it ; roaring like claps of thunder ; which made us run
out of doors to see what was coming upon us. We ran to look at
the Gill ; and we directed our sights (by the noise that it made) the
right way ; and the frightfulness of the appearance at the very first
sight, which was when about the middle of the pasture, made us run
for our lives ; and we got no further than from the yet (j. e. gate) to
the sycamore trees, before the stable, peat-house, and all the calf-
parrack (i. e. paddock) and cow-parrack, was in a heap of the most
shocking ruins that ever your eyes beheld. I believe from the first
sight of it, when it was coming, till all was overturned, was less than
the quarter of a minute's time. It has brought rocks down past the
middle of the houme, which had gone through the peat-house and
stable, that I think three or four yoke of oxen could not be able to
move. The poor old horse was crushed to pieces in a moment.
Nothing but the good providence of God has preserved us from pe-
rishing ; for it's amazing to think how the bEirn stood the violence of
the shock. The waters run round our dwelling house, broke down
the garden wall, and continued running through it till next day in
the morning ; so that it's become a bed of sand. It was about 1 1
o'clock when this happened, and we went from place to place, not
knowing where to be safe, expecting every moment more of the like
nature ; which accordingly happened ; for I think in the space of two
hours the face of things was so changed that one scarcely could have
known them. For they came down almost every slack*, carrying all
the walls before them ; so that we were obliged to run from one
place to another to escape their fury, which was with difliculty : for
it continued raining extremely, that we were wet to the naked skin,
not daring to come in any house. And it drawing towards night, we
resolved to make an attempt to get to brother John's, and accordingly
set forwards, and got up at our pasture head on to the moor, and
with difficulty got over Harbourgill, and so forwards to the Mun-
keybeck. But we knew that the bridge was broke down, so that we
must be obliged to pass it somewhere on the moor, and we waded
through the water and snow till we were almost spent in extreme
wet and fatigue ; and at last got over a little below where our peat-
fell is (tho' with very great, hazard of our lives), at last, my poor
old Father and Betty being almost quite spent, he having only one
* Slack (coowi or hollow in the hill-side).
238
shoe on one foot the greatest part of that time. Then when we were
got over, it gave us some fresh encouragements, and we arrived at
Bro' Johns just before it was dark, where we were thankful to see
the faces of one another in a phice of more safety. We went three
nights successively to Bro' John's to lodge, not daring to stay about
the old place. Old Francis Swinbank [rect. Swithinbank] and
Thomas Stockdale's whole family perished in a moment about the
same time that the thing happened with us, being seven in number.
Likewise John Burton, Stone House, had a barn swept away and a
cow killed.
" I hope these few broken hints will be excused, for I am not very
good at writing at this time, all being so in confusion. Sr. greatly
desires you would come to see us as soon as well can. For our love
is very much towards you. You perhaps may think 1 have out-
stretched, but if you please to come your eyes will convince you to
the contrary. For I have not told you one half. So shall conclude
your very loving Brother,
" Thomas Thistlethwaite."
" Betty's kind love is to you both, but Sr. in particular,"
November 27, 1865.
Mr. J. W. Clark, Trinity College, read a paper upon the Rib of a
Whale found by some fishermen near Sherringham (four miles N.W.
of Cromer). It was discovered after a high tide, which caused a
fall of the cliff; it was reported to have been imbedded in drift gravel.
Mr. Clark stated he had compared it with the rib of Physalus, with
which it did not agree ; for the tubercle and head of the rib were
very much wider apart in the fossil specimen than in Physalus. It
resembled Balcenoptera more nearly, and still more closely Balcena
mysticetus. It was probably the fourth or fifth rib of the left side.
He remarked that a few months since some cervical vertebrae had
been found at Plymouth which probably belonged to Balcena
Biscayensis, a whale which was nearly extirpated in the sixteenth
century. It was, however, still occasionally met with, a cow and
cul) being not long since seen near S. Sebastian, and the latter cap-
tured. He could not positively refer the rib exhibited to any known
species.
Professor Sedgwick remarked that large cetacean vertebrae had
been found at Landbeach, Cambridgeshire, and in the Crag of Nor-
folk and Suffolk. This was the largest rib that he had ever seen;
but he had great doubt, from its general appearance, whether it
could have come out of the gravel.
Mr. Newton remarked that our knowledge of whales had of late
been greatly enlarged ; for of the bodies which had been stranded
so many new species and even new forms had been observed, that
hardly two had been found to be identical.
239
Professor Miller exhibited two new forms of heliotrope, explaining
at the same time the difficulties in signalling, which they were in-
tended to overcome, commenting upon the relative merits of those
invented by Gauss, Steinheil, and others, and explaining the spe-
cial advantages of the two which he exhibited.
Professor Miller also communicated a Supplement to the Crystal-
lographic method of Grassmann.
Mr. G. F. Browne, St. Catherine's College, communicated some
Notes upon some Ice-caves explored during the summer of 1865.
Two of these he had visited during the previous summer, and he
found that there was a somewhat greater quantity of snow in the
caverns than there had formerly been. In the first cave he again
examined a pit in the ice about 70 feet deep, but, owing to the dan-
gerous condition of the ice, was unable to descend into it. In the
second cave he had again cut through a curtain of ice into an icy
tunnel ; but this year the diameter of the tunnel was so much smaller
that he was unable to descend it, although provided with ropes for
the purpose. He described some flies found inside the tunnel. The
third cave had not previously been explored ; it was an oval in shape,
with a level floor of ice. He had descended for about 12 feet
between the ice and the rock, and there found a narrow tunnel which
appeared to lead to a subglacial reservoir containing water. He
ascertained that the ice was at least 24 feet thick ; but it was impos-
sible to descend the tunnel.
Mr. Bonney, who had accompanied Mr. Browne, made some ob-
servations on the general character of the country, expressing his
opinion that the glacieres were formed by the accumulation of snow
in suitable fissures ; and remarking that the prismatic structure of
the ice noticed last year by Mr. Browne was very conspicuous.
INDEX OF NAMES.
VOL. I.
N.B. — The larger figures indicate separate papers; the smaller,
incidental references.
Adams, J. C. F.K.S. 119, 192
Airy, G. B. 27, 62, 205, 218
Akin, Dr, 227
Amsler, Prof. 192
Angstrom, Prof. 1 36
Babington, C. C. 127, 234
Babington, Prof. Churcliill, 234
Barrett, L. 168, 218
Bashforth, Eev. F. 173, 182
Baxter, H. F. 200
Bezont, M. R3
Blakeslej, Eev. J. W. 218
Bonney, Rev. T. G. 239
Browne, Eev. G. F. 230, 239
Calvert, Eev. F. 74
Candy, F. J. 182
Camot, M. 66
Catton, A. E. 230, 235
Cauchy, M. 144
Cayley, Prof. 228, 228, 232, 235
ChaUis, Prof. 26, 27, 73, 75, 75, 120,
122, 123, 132, 133, 134, 146, 179,
203, 204, 216, 219, 226, 227
Clark, Prof. M.D. 148
Clark, J. W. 238
Clark, W. G. 204, 223, 224
Clifton, Prof. 232
Cox, Homersham, 88
Decher, Prof. 192
De Morgan, Prof. 1, 6, 13, 29, 64, 90,
106, 121, 123, 136, 145, 155, 172,
190, 194, 194, 208, 222, 228, 232
Denison, E. B. 77, 121
De Senarmont, 120, 136
Dobson, 127
Donaldson, Eev. J. W. D.D. 175, 199,
206, 217
Eamshaw, Eev. S. 14, 25
Ellis, E. L. 5, 63, 147
Fisher, Prof. 132, 146
Fisher, Eev. 0. 147, 158, 194
Forbes, Prof. 4
Ganss, F. 136
Godfray, H. 203
Goodwin, Eev. H. 15, 19, 28
Graham, Prof. 200
Grassmann, Prof. 234, 239
Harrison, A. A. 169
Hayward, R. B. 166
Hegel, 84
Hopkins, W. F.E.S. 2, 9, 116, 126,
148, 149, 181, 209, 217
Humphry, G. M. M.D. 177, 181, 206,
226, 227, 230
Kemp, G. M.D. 21
Kingsley, Eev. W. T. 117, 145, 182
Lesearbault, M. 219
LionviUe, M. 136
Liveing, Prof. 227, 234
Mc Coy, F. 64
Macleod, H. D. 226, 227
Maxwell, J. Clerk, 134, 149, 160, 163,
173
MiUer, Prof. 116, 117, 119, 136, 177,
192, 203, 206, 209, 218, 226, 234, 239
Mitscherlich, 136
Monge, M. 136
Moon, Eev. E. 75, 79
Morgan, see De Morgan
Munro, Eev. H. A. J. 218
Newton, Alfred, 232, 238
O'Brien, Rev. M. 33, 38, 56
Paget, G. E. M.D. 29, 148, 200, 224
Phear, J. B. 133
Porro, M. 206
Potter, Prof. 20
Power, Rev. J. 179
Pritchard, Rev. C. 122
Regnanlt, M. 66
Rdhrs, J. H. 115, 223, 228
Sarrus, M. 109
Sedgwick, Prof. 158, 187, 218, 223,
236, 238
Seeley, H. 226, 228, 228, 230, 234, 234
Selwyn, Prof. 230
Steele, W. 70
Stokes, G. G. 16, 19, 35, 37, 61, 78,
82, 83, 85, 95, 104, 110, 111, 115,
117, 119, 120, 122, 132, 181, 209
Thompson, Prof. W. H. 191
Thomson, Prof. W. 66
Todhunter, I. 227, 234
Tozer, J. 2
Warburton, H. M.P. 43, 145, 173
Wedgwood, 127
242
Wetli, Prof. 192 Wiedemann, 136
Whewell, W. D.D. 4, 74, 81, 84, 89, WUlis, Prof. 148
99, 100, 103, 110, 148, 149, 158, 176,
203, 204, 217, 226 Young, Prof. 36
SUBJECTS.
Aberration of light, 19
Absolute thermometric scale, 66
Adjustments of a transit, 123
Algebra, foundation of, III. 1
IV. 13
Algebraic Equations, Geom. repre-
sentation of their roots, 28
Analysis, continuity in, 36
Angle, infinite, 14
Antithesis of Philosophy, fundamen-
tal, 4, 74
Arbitrary constants, discontinuity of,
181
Arctic Eegions, geological discoveries
in, 219
Argument, origin and meaning of the
word, 217
Aristotle's account of induction, 89
Astronomical instruments, construc-
tion of, 206
Athenian trireme, structure of, 175
Atwood's machine, 148
Aurora Borealis, 20
Avalanche in 1752, 236
Axis of a zone, direction of, 203
Balance, knife edges of, 119
Banking, theory of, 227
Barrow, and his academical times,
203, 204
Biela's double comet, 120
Body and Space, knowledge of, 97
Bones, growth of, 226
Bows, motion of, 223
Calorimeter, a new, 217
Camera lucida prisms, 218
Capillary action, theories of, tested,
182
Carboniferous period, fossil fish of,
64
Caustic, intensity of light near, 62
Chemical compounds, 230
Clock escapements, 77, 121
Clouds, height of, 119
Coinage of England, 234
Colours, experiments on, 149
„ of thick plates, 110
Combinations, &c. of numbers, 43
Comet, Biela's, 120
„ Donati's, 204
„ Klinkerfue's, 132
Composition of forces, principles of,
208
Conductivity of substances, 181
Consonances, imperfect, 190
Continuity, principle of, 36
Crystals of an anorthic system, 203
,, of the oblique system, 136
Crystallized minerals, 116
Crystallography, sphere in, 209
„ investigation of, 234,
239
Cubic cones and curves, theory of, 228
Curve, equation to, 81, 83, 100
Curves, on singular points of, 155
Cyclones, theory of, 127
Dent, geology of the valley of, 236
Development of vertebral system, 132,
146
Diffraction, dynamical theory of, 85
Dispersion of light, 119
Divergent series, &c., 6, 181
Dorsetshire, deep pits in, origin of,
194
,, Purbeck strata of, 147
Earth, interior temperature of, 126,
see 181
,, superficial do. 116
,, changes in do. 148
Earth and other planets, external
temperature of, 149
Earthquake in Switzerland, 158
Eclipse of the Sun, Mar. 15, 1858, 203
July 18, 1860, 226
Elastic beams, impact on, 88
,, medium, equation for motion
of, 56
,, rods, motion of, 223
,, solids, friction of, 16
Electricity, origin of, 227
Elliptic analyzer, an. 111
Entozoa, discoveries respecting, 148
Equality, sign of, 229
Equation, algebraic, proof of root of,
194, 205, 209, 218
,, relating to bridges, 83
,, to a curve, 81, 83, 100
,, differential, of the first
order, 101
„ „ solution of, 172
,, ,, theory of, 136
243
Equation for an elastic medium, 56
Equations, partial differential, 64
„ numerical, 79
Equilibrium of forces, theorem on, 235
Erratic blocks, transport of, 9
Escapements, clock, 77, 121
Euclid, geometry of, 127
Eye, affected by malformation, 27
FaU, tendency to, 148
Fen-land, gravel and drift of, 234
Fluids, effect of friction on pendu-
Imns, 104
,, in motion, friction of, 16
Force, Faraday's Lines of, 160
Forces, composition of, 209
,, equilibrium of, 235
Formic acid, synthesis of, 235
Fossil birds, 228
„ fish, 64
Fossils, succession of, 228
Fusion, temperature of, under pres-
sure, 126
Geology of the Lake District, 187
Suffolk, 133
the VaUey of Dent, 236
Geological discoveries in the Arctic
Eegions, 219
Geometry, abstract, 232
„ &c. new notation for, 33
„ symbolical system of, 38
of Euclid, 127
Gipping, vaUey of, 133
Glaciers, on the motion of, 2, 209
Gnomonic projection of sphere, 209
Goniometer, reflective, 117
,, ,, substitute for,
203
Gravity, variation of, 82
Great-circle sailing, chart for, 203
Greek, ancient and modern, 223
,, accentuation of, 224
Green sand, Cambridge, 234
,, ,, upper, fossil birds of, 228
Growth of bones, 226
,, ,, the jaw, experiments on, 227
Gun cotton, 234
Gunpowder, effects of, 228
Haidinger's brushes, 117, 179
Halos, parhelia, &c. 122
Heat, theory of, 169
HeUotrope, new portable, 218, 239
Human frame, on the proportions of,
181
Hypotheses, transformation of, 110
Ice-caverns, 230, 239
Induction, Aristotle on, 89
Infinite angle, values of sine and
cosine, 14
Infinity, on, 229
Instruments, surveying and astrono-
mical, 206
Integral calculus, on some points ofj
106
Integrals, numerical calculation of, 95
Integration of diff. equations, 64
Internal friction of fluids, ** vanish together.
Also if X, y be the rectangular coordinates of P, the axis
of X being parallel to B G,
dx _dx ds _ r
d ds' d) '
. P _ f d.sm(f> _ 1 . 1 + sin a . sin ^ ^
**T* J 1 — sin^a.sin*^ 2 sin a ° 1 — sina.sin ^ '
p _ r sin (f}d(f> _ C d. cos ^
T ] 1 — sin* a . sin* ^ ~ j cos' a + sin*a . cos' ^
. tan"* (tan a cos ^) ;
so
sm a . cos a
no constants are required if x vanishes when ^ = 0, and y when
^ 2
230
Therefore on eliminating (f>,
p • p ■
-.sina.x , _ — .sma.x
+ 2 sec a. COS (sin a cos a.- .3/] .
In the second case, that is, when p is less than ^ av^, assume
(TV
p _ f ^ • ^^^ ^ — *^^ ^ 1 *^^ ^ + ^^^ ^
then - . s
C d . tan (f> _ tan p ,
"Jl-cot'^/3.tan''' 2^'^^^-
?-=/r
cof* /3 . tan'-* 2 ' ^ * tan /8 ~ tan (^
_ tan ^8 , sin (yS + ^)
~"~2~' ^^•'^in(^'-<^)'
cos (f) . dS sin ;3 , sin 8 + sin
cosec^ /3 . sin*^ ^ 2 ' ° * sin ^ ~ sin ^
. tan^=^^
sm/3 , 2
tan-^
p _ r sin ^.d(f) _ r d.cos(f)
t'" J 1 — cosec^ yS . sin^ ^ J cosec'' /3 . cos^ ^ — cot* /3
= tan' /3 f ^-^Qs *^ = sin'^ 1 cos + cos /3
jl — sec'^/S. cos'^ 2cos;S' ^*cos^~cos/8
after eliminating ^, the rectangular equation is found to be
cosj3 p _ 1 p cos/3 p 1^ p
+ gsin2^*T'^ sin/3" t'^ + g sm*/3*T"^ sin/3 't**
tan yS = =
2COS/3 p
4.gsin»/3 •t'''_1
In the case when^ = ^ (xv^,
2
p f ^ X -'^ X n
~ .s= jsec'**