584 
NATORE 
([OcToBER 12, 1899 
cation by Dalzell in 1861 of his ‘* Flora of Bombay.” It is im- 
possible in a brief review like the present to mention the names 
of all the workers who, in various parts of the gradually ex- 
tending Indian Empire, added to our knowledge of its botanical 
wealth. It must suffice to mention the names of a few of the 
chief, such as Hardwicke, Madden, Munro, Edgeworth, Lance 
and Vicary, who collected and observed in Northern India, and 
who all, except the two last mentioned, also published botanical 
papers and pamphlets of more or less importance ; Jenkins, 
Masters, Mack, Simons and Oldham, who all collected ex- 
tensively in Assam; Hofmeister, who accompanied Prince 
Waldemar of Prussia, and whose collections form the basis of 
the fine work by Klotsch and Garcke (7ezs. Pr. Wald.) ; Norris, 
Prince, Lobb and Cuming, whose labours were in Penang and 
Malacca ; and last, but not least, Strachey and Winterbottom, 
whose large and valuable collections, amounting to about 2000 
species, were made during 1848 to 1850 in the higher ranges of 
the Kamaon and Gharwal Himalaya, and in the adjacent parts of 
Tibet. In referring to the latter classic Herbarium, Sir Joseph 
Hooker remarks that it is ‘‘ the most valuable for its size that 
has ever been distributed from India.” General Strachey is the 
only one who survives of the splendid band of collectors whom 
I have mentioned. I cannot conclude this brief account of the 
botanical labours of our first period without mentioning one 
more book, and that is the ‘‘ Hortus Calcuttensis” of Voigt. 
Under the form of a list, this excellent work, published in 1845, 
contains a great deal of information about the plants growing 
near Calcutta, either wild or in fields and gardens. It is strong 
in vernacular names and vegetable economics. 
(Zo be continued. ) 
MATHEMATICS AT THE BRITISH 
ASSOCIATION. 
HE visit of the French Association to Dover necessitated 
some departures from the usual programme of the British 
Association week, and the mathematical meeting was held this 
year on Monday, September 18. Prof. Forsyth, of Cambridge, 
presided over a well-filled room, 
The session opened with the formal communication of two 
reports of committees: the first, drawn .up by Prof. Karl 
Pearson, and practically forming a continuation of a previous 
report, contains a set of tables of certain functions connected 
with the integral 
G(r, v)= | ” sin "e"*a0, 
0 
for integral values of 7 from 1 to 50, and for values of v at cer- 
tain intervals from o to 1. These functions are of importance 
in certain statistical problems. 
The second report consists substantially of the new ‘“ Canon 
Arithmeticus” which Lieut.-Colonel Cunningham has _pre- 
pared ; the Association has made a grant for publishing the 
tables as a separate volume (they cannot well be fitted into the 
comparatively small page of the B.A. Report), and it is to be 
hoped that before long they will become generally available for 
workers in the Theory of Numbers. 
The first of the papers was read by Dr. Francis Galton, on 
“The Median Estimate.” Dr. Galton proposes to substitute a 
scientific method for the very unsatisfactory ways in which the 
collective opinion of committees and assemblies of various kinds 
is ascertained, in respect to the most suitable amount of money 
to be granted for any particular purpose. How is that medium 
amount to be ascertained which is the fairest compromise 
between many different opinions? An average value—z.e. the 
arithmetic mean of the different estimates—may greatly mislead, 
because a single voter is able to produce an effect far beyond his 
due share by writing down an unreasonably large or unreason- 
ably small sum. Again, few persons know what they want with 
sufficient clearness to enable them to express it in numerical 
terms, from-which alone an average may be derived; much 
deeper thought-searching is needed to enable a man to make such 
a precise affirmation as that ‘‘in my opinion the bonus to be 
given should be 8o/.,” than to enable him to say ‘‘I do not think 
he deserves so much as 100/., certainly not more than 100/.”’ 
Dr. Galton’s plan for discovering the medium of the various 
sums desired by the several voters is to specify any two reason- 
able amounts A and B, and to find what percentage a of voters 
think that the sum ought to be less than A, and what per- 
centage 4 vote for less than B. It may now be assumed that 
NO. 1563, VOL. 60] 
the estimates will be distributed on either side of their (unknown) 
median m, with an (unknown) quartile g, in approximate 
accordance with the normal law of frequency of error ; and thus 
(using the table of centiles given in the author’s ‘‘ Natural 
Inheritance *’) the required median value can be found, 
This was followed by a paper ‘‘ On a system of invariants for 
parallel configurations in space,” by Prof. Forsyth. The 
process followed by the author is one in which English mathe- 
maticians have always excelled—namely, the deduction of 
difficult analytical results from simple geometrical considerations. 
Prof. Forsyth’s final formulz may be regarded as invariantive 
relations between certain definite integrals ; the way in which 
he finds them is as follows :— 
Consider any plane curve; if we suppose a circle of constant 
size to roll on the curve, its envelope will be another curve, 
which is said to be farall/el to the original one. If now L 
be the length and A the area of a curve, it is found that the 
quantity A — 1 L? has the same value for the parallel as for the 
47 
original curve ; in other words, 
I 
ae 
A= reo 
is zzvardantive for parallel curves. Similarly in space of three 
dimensions, the envelope of a sphere of fixed size which rolls 
on a given surface is another parad/e/ surface ; and if V be the 
volume contained by a surface, S its superficial area, and L 
twice the surface-aggregate of the mean of the curvatures at any 
point, then it is found that the quantities 
I 
Sey fe 
To2m/ 
Lyte _i 
Se ea and V g,LS+ 
are invariantive for all parallel surfaces. 
Similar results hold for space of 7 dimensions. At the end 
of the paper the expressions obtained are shown to be 
connected with the ordinary invariant-theory of binary forms. 
The next paper, read by Prof. Everett, was concerned with 
“The Notation of the Calculus of Differences.” In conjunction 
with the ordinary symbol A, defined by 
DBIn=In41—Ins 
Prof. Everett employs another symbol 6, defined by 
dyn =In-In-1> 
so that 
8=A/, +4. 
The use of 5 simplifies some of the well-known formulz of 
the calculus of finite differences. 
Prof. A. C. Dixon, of Galway, followed, with a paper ‘‘On 
the Partial Differential Equation of the Second Order.” Let 
z be the dependent, and x and y the independent, variables ; 
and with the usual notation, let 
Oz Oz 02" 02" Oz" 
= = a = a — 
ax «oy aoe axdy, ay, 
and consider the differential equation 
T(x, Is 25 Pr Ws 7s Sy th=0, 
This may be supposed solved by using two more relations 
= 0h DSF 
among the quantities «, y, 2, 2, 7, 7 5, ¢, to give values of 7, s, 2, 
which, when substituted in 
dz=pax + qdy, adp=rdx+say, dg=six+tdy, 
render these three equations integrable. This will not be 
possible, of course, unless the expressions x, 7, fulfil certain 
conditions. Prof. Dixon considers the case in which z# can be 
so determined that 7 is only subjected to one condition, and 
finds that then az is a linear combination of the differentia} 
expressions used in Hamburger’s method of solution. If such 
a function # can be found, the system /=0, «=a, will have a 
series of solutions depending onan arbitrary function of one 
variable, and involving two further arbitrary constants. 
The next paper, ‘‘On the Fundamental Differential Equa- 
tions of Geometry,” was read by Dr. Irving Stringham, of the 
University of California. Dr. Stringham derives the analytical 
formule for non-Euclidian Geometry by following a procedure 
indicated by Feye St. Marie, and later discussed in Killing’s 
‘* Nicht-Euclidischen Raumformen.” Within an infinitesimal 
domain in non-Euclidian space, the propositions of Euclidian 
