paper on M. Jacobi’s Theory of Elliptic Functions. 359 
he says, “ For these forms the equations (3.) and (4.)* are in their 
simplest forms; their second members vanish when wv = 0, 2, 
4., &c., but never between these values. Consequently, while 
w increases by 2, i increases by 2 H, neither more nor less, if 
Ma H when its amplitude is = Whilst, therefore, w from 
0 becomes a, M from 0 becomes H. Let them have these 
values in (3.), and we obtainsa H=1= + Csawsa3wu. 
sa(2n—1)a, age +saw..sa(2n—1). This then 
1 
is the general form of ok and M.Jacobi’s denominator can- 
not be true, except in those cases in which it is reducible to 
it.” I have quoted this verbatim, or I should probably have 
misrepresented it, for I am utterly unable to see the force of 
it. For anything I can see to the contrary, when zu increases 
by Zo, ie might increase by 2p H + 2p'H! —1 (as Mr.° 
Bronwin expresses it), or , instead of being equal to M H, 
might be equal to M (pH + p! H! W —1), p, p! being inte- 
gers to be determined. Hence when wu from 0 becomes a, 
u 
M ee 
the first side is of one of the forms + 1,0, +o VW —1, 
from 0 becomes pH + p! H! / — 1. Substituting in (3.) 
1 
k 
+ iy according to the forms of p and p!. The second side 
contains the factor saw, which is likewise of one of the 
1 , 
forms + 1,0, +o VY —1, toy according to the form of w. 
Thus it may happen that C, instead of having Mr. Bronwin’s 
ot fel ’ 0 . 
value, is given by an equation C = oF C = &, or that its 
value cannot be determined by this process, and may, on the 
contrary, be determined by M. Jacobi’s formula. In the re- 
maining case excepted from Mr. Bronwin’s reasoning, he 
certainly shows that M. Jacobi’s formula: are not in their 
simplest form, but by no means that they fail; on the con- 
trary he rather confirms them, I may just mention the ar- 
gument of a note in the Cambridge Mathematical Journal, 
* I have not written down here the equation (4.), which is not neces- 
sary for my present purpose. 
