ON THE THEORY OF POINT-GROUPS, 29 
ganetion Of Z,,;,(.-. —ap..-—al™) + ule +... uP” +e...) con- 
gruent to the sum of the values of all its p zeros ; now the m points 
Ep-1+ + + €p-m are m of these zeros, since when ¢,,;=2,1 . . . 2m in 
L 
turn, we get a $-function whose arguments are of the form of (1) ; let the 
p—m other zeros be n, . ~~ %)-m then the above congruence, when 
identical terms are removed from each side, becomes 
eee ale, 2 2 Ja (2s ob ie Fal a yt 
But again, by considering (2) as a function of z,_,, we find that, 3 being 
an even function, (...ai?t?+... +aPo™*P—upP— 1... —uPe-m—e,... 
is congruent to the sum of its values at its p zeros; and that m+1 of 
these zeros are 741 - - - Mp—m+1, Since when 2, ;=a?t” , . .a?-™+*” in turn 
we get a 3-function whose arguments are of the form (1) ; now let the 
p—m-—1 other zeros be ¢,; . . « €)~ i, then the congruence which holds is 
Ret le sage er ee Jw Os ce WOES ome is toh 
We have thus shown that (e, . . . e,) = (ul? ... a) and also to 
(—u ... —u”), d.e., that the point-groups composed of the p 7’s and 
those composed of the p—2 «’s are congruent to each other ; and, more- 
over, that when m of the points » are arbitrary (p—m being uniquely 
determined), then m—1 of the points « are arbitrary (since p—m—1 of 
the p —2 are determinate). And it is easily seen that this relationship is 
reversible, z.¢., that if m—1 of the «’s are arbitrarily chosen, then a of the 
ys can be arbitrarily chosen. The 2»—2 zeros of a o-function have thus 
been divided into two point-groups, containing p and p—2 points 
respectively, and it has been shown that if m of the p points can be 
arbitrarily chosen, then m—1 of the p—2 points are also arbitrary, and 
vice versa. 
A precisely similar line of argument applied to the conclusions of § 23 
shows that the 2»—2 zeros of a ¢ function may also be divided into two 
point-groups, each consisting of p—1 points, and that then, if m of one 
point-group are arbitrary, m of the other are also arbitrary. 
The connection with the Riemann-Roch theorem is at once evident ; 
for if in the first case dealt with above we assume that a rational function 
becomes infinite at p points, then, if m of these p points are arbitrary, the 
function has m arbitrary constants, and therefore, since by the Riemann- 
Roch theorem m=p—p+1+q, g=m—1, i.e., m—1, different o functions 
can be drawn through them, which agrees with the number of «’s which 
have been shown to be arbitrary ; and, conversely, if the number of 
infinities is p—2, and if m—1 only are arbitrary, the Riemann-Roch 
theorem shows that m—l=p—2—p+1+4, ie., that g=m, which agree 
with the number of 7’s which may be chosen arbitrarily. 
Magnetic Observations at Falmouth Observatory.—Report of the 
Committee, consisting of Sir W. H. PREEcE (Chairman), Dr. 
R. T. Guazeproox (Secretary), Professor W. G. Apams, 
Captain Crea, Mr. W. L. Fox, Principal Sir Anraur Ricker, 
and Professor A. SCHUSTER, appointed to co-operate with the 
Conumittee of the Falmouth Observatory in their Magnetic Obser- 
vations. 
THE grant voted by the Association last year has been expended in 
carrying on the Magnetic Observations at Falmouth Observatory. 
. The apparatus at the Observatory was inspected by Mr. T. W. Baker 
