16 
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shall treat this problem only under certain restrictions, and Ishall 
confine myself in the first place to lines in which the ratios of 
the increments dx of the respective variables vary continu- 
ously. We may then conceive these lines broken up into 
elements, within which the ratios of the quaniities dv may 
be regarded as constant; and the problem is then reduced 
to establishing for each point a general expression for the 
linear element ds starting from that point, an expression 
which will thus contain the quantities + and the quantities 
dx. 1 shall suppose, secondly, that the length of the 
inear element, to the first order, is unaltered when all the 
points of this element undergo the same infinitesimal dis- 
placement, which implies at the same time that if all the quan- 
tities dr are increased in the same ratio, the linear element will 
vary also in the same ratio. On these suppositions, the linear 
element may be any homogeneous function of the first degree of 
the quantities @x, which is unchanged when we change the signs 
of all the 2x, and in which the arbitrary constants are continuous 
functions of the quantities «. To find the simplest cases, I shall 
seek first an expression for manifoldnesses of #—1 dimensions 
which are everywhere equidistant from the origin of the linear 
element ; that is, I shall seek a continuous function of position 
whose values distinguish them from one another. In going 
outwards from the origin, this must either increase in all direc- 
tions or decrease inall directions ; I assume that it increases in 
all directions, and therefore has a minimum at that point. Tf, 
then, the first and second differential coefficients of this function 
are finite, its first differential must vanish, and the second diffe- 
rential cannot become negative ; I assume that it is always posi- 
tive. This differential expression, then, of the second order 
remains constant when ds remains constant, and increases in the 
duplicate ratio when the dx, and therefore also ds, increase in 
the same ratio; it must therefore be ds* multiplied by a con- 
stant, and consequently ¢@s is the square root of an always posi- 
tive integral homogeneous function of the second order of the 
quantities dx, in which the coefficients are continuous functions 
of the quantities «, For Space, when the position of points is 
expressed by rectilinear co-ordinates, ds=./3(dx)°; Space is 
therefore included in this simplest case. The next case in sim- 
p icity includes those manifoldnesses in which the 1 n= element 
may be expressed as the fourth root of a quartic differential ex- 
pression. ‘The investigation of this more ge neral kind would 
require no really different principles, but would take considerable 
time and throw little new light on the theory of space, especially 
as the results cannot be geometrically expressed ; I restrict my- 
self, therefore, to those manifoldnesses in which the line-element 
is expressed as the square root of a quadric differential expres- 
sion. Such an expression we can transform into another similar 
one if we substitute for the 7 independent variables functions of 
2 new independent variables. In this way, however, we cannot 
transform any expression into any other ; since the expression 
u+t 
contains 7 coefficients which are arbitrary functions of 
the independent variables ; now by the introduction of new 
variables we can oaly satisfy 7 conditions, and therefore make 
no more than # of the coefficients equal to given quantities. 
The remaining x» “— T are then entirely determined by the 
nature of the continuum to be represented, and consequently 
n-1{ : 
n . functions of positions are required for the determina- 
tion of its measure-relations. Manifoldnesses in which, as in 
the Plane and in Space, the line-element may be reduced 
to the form «/ Sx, are therefore onlya particular case of the 
manifoldnesses to be here investigated ; they require a special 
name, and therefore these manifoldnesses in which the square of 
the line-element may be expressed as the sum of the squares of 
complete differentials I will call #a¢. In order now to review 
the true varieties of all the continua which may be represented 
in the assumed form, it is necessary to get rid of difficulties 
arising from the mode of representation, which is accomplished 
by choosing the variables in accordance with a certain principle. 
§. 2.—For this purpose let us imagine that from any given 
point the system of shortest lines going out from it is constructed ; 
the position of an arbitrary point may then be determined by the 
initial direction of the geodesic in which it lies, and by its dis- 
tance measured along that line from the origin, It can therefore 
be expressed in terms of the ratios @v of the quantities dv in 
this geodesic, and of the length s of this line, Let us intro- 
NATURE 
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on 
duce now instead of the @% linear functions dx of them, such 
that the initial value of the square of the line-element shall 
equal the sum of the squares of these expressions, so that the 
independent variables are now the length s and the ratios of 
the quantities dx. Lastly, take instead of the d x quantities 
4X, X%yXz...%, proportional to them, but such that the sum of their 
squares = When we introduce these quantities, the square 
of the line-element is = @.2* for infinitesimal values of the «, but 
the term of next order in it is equal to a homogeneous function’ 
of the second order of the x “—! 
quantities (x, dx,—4%,dx,), 
x, @%4—-xX3d@x,)... an infinitesimal, therefore, of the fourth 
I 3 3 4 
order ; so that we obtain a finite quantity on dividing this by the 
square of the infinitesimal triangle, whose vertices are (0, 0, 0,...), 
(x, %_%y...), (€x,, Axy, AX y,...). This quantity retains the 
same value so long as the x and the ¢ are included in the same 
binary linear form, or so long as the two geodesics from 0 to x 
and from o to dx remain in the same surface-element ; it depends 
therefore only on place and direction. It is obviously zero when 
the manifold represented is flat, ze. when the squared line- 
element is reducible to S@«*, and may therefore be regarded as 
the measure of the deviation of the manifoldness from flatness at 
the given point in the given surface-direction, Multiplied by 
— it becomes equal to the quantity which Privy-councillor 
Gauss has called the total curvature of a surface. For the 
determination of the measure-relations of a manifoldness capable 
Of representation in the assumed form we found that ne 
place-functions were necessary ; if, therefore, the curvature at 
i! 
each point in »” surface-directions is given, the measure- 
relations of the continuum may be determined from them— 
provided there be no identical relations among these values, 
which in fact, to speak generally, is not the case, In this way 
the measure-relations of a manifoldness in which the line-element 
is the square root of a quadric differential may be expressed in 
a manner wholly independent of the choice of independent vari- 
ables. A method entirely similar may for this purpose be ap- 
plied also to the manifoidness in which the line element has a 
less simple expression, ¢g., the fourth root of a quartic 
differential. In this case the line-element, generally speaking, 
is no longer reducible to the form of the square root of a sum of 
squares, and therefore the deviation from flatness in the squared 
line-element is an infinitesimal of the second order, while in 
those manifoldnesses it was of the fourth order. This property 
of the last-named continua may thus be called flatness of the 
smallest parts. The most important property of these continua 
for our present purpose, for whose sake alone they are here in- 
vestigated, is that the relations of the twofold ones may be geo- 
metrically represented by surfaces, and of the morefold ones may 
be reduced to those of the surfaces included in them; which now 
requires a short further discussion. , 
§ 3.—In the idea of surfaces, together with the intrinsic mea- 
sure-relations in which only the length of lines on the surlaces is 
considered, there is always mixed up the position of points lying 
out of the surface. We may, however, abstract from external 
relations if we consider such deformations as leave unaltered the 
length of lises—z.e. if we regard the surface as bent in any way 
without stretching, and treat a1 surfaces so related to each other 
as equivalent, hus, for example, any cylindric or conical sur- 
face counts as equivalent to a plane, since it may be made out of 
one by mere bending, in which the intrinsic measure-relations 
remain, and all theorems about a plane—therefore the whole of 
planimetry—retain their validity. On the other hand they 
count as essentially different from the sphere, which can- 
not be changed into a plane without stretching. According 
to our previous investigation the intrinsic measure-relations of a 
twofold extent in which the line-element may be expressed as 
the square root of a quadric differential, which is the case with 
surfaces, are characterised by the total curvature. Now this 
quantity in the case of surfaces is capable of a visible interpre- 
tation, viz, it is the product of the two curvatures of the surface, 
or multiplied by the area of a smatl geodesic triigle, it is equal 
to the spherical excess of the same. The first cefinition assumes 
the proposition that the product of the two radii of curvaiure is 
unaltered by mere bending; the second, that in the same 
place the area of a small triangle is proportional to its spherical 
excess. To give an intelligible meaning to the curvature of an 
n-fold extent at a given point and in a given surface-direction 
through it, we must start from the fact that a geodesic proceeding 
[May ty 1873. a 
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