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XVII. On a Class of Identical Relations in the Theory of Elliptic Functions. 
By J. W. L. Glaisher, M.A., Fellow of Trinity College , Cambridge. 
Communicated by James Glaisher, F.R.S. 
Received November 23, 1874, — Read January 14, 1875. 
§ 1. The object of the present paper is to notice certain forms into which the series for 
the primary elliptic functions admit of being thrown, and to discuss the identical rela- 
tions to which they give rise. These latter, it will be shown, may be obtained directly 
by the aid of Fourier’s theorem, or in a less straightforward manner by ordinary 
algebra. 
§ 2. Whenever we have a periodic function of x, say \px, such that yJ/x=ip(x-}-yj), it 
is well known that we may assume, for all values of x, 
i \ i a 2vx . a 4«r 
V/ar=A 0 +A 1 cos \-A 2 cos \- &c. 
/J. - fL 
-j-Bi sm — +B 2 sin f- &c. ; 
p, y. 
and if ypx be even, so that ypx=ip(—x), then B„ B 2 , &c. all vanish; while if \j/x is 
uneven, so that \px= — \p(—x), A 0 , A 1? &c. vanish. If ypx is such that ■tyx=-$(x-\-p), 
then we have 
• a 7rx . 3irx p 
■d/x= A. COS kA„ cos — 4- &c. 
T l y. ' 3 y. 1 
. ntx „ 3ttx „ 
=Bj sin — + B 3 sm — + &c., 
according as -tyx is even or uneven. 
But there is another totally different form in which ipx may generally be exhibited, 
viz. 
'px=<px+<p(x—{A)-±-<p(x-hft)-l-<p(x—2y,)-l-<p(x+2p)-i- &c. 
or 
zspx—p(x—fi)—p(a+f6)+p(x— 2fA)+p(x+2jt)— &c., 
according as yp(xA-g')=4'X or = — ipx. 
The sine and cosine cannot be so expressed, but the other primary circular functions 
do admit of this form, as, ex. gr., in the formulae 
cot tT— — -J- ] -4 7. -J- . 0 + &c., 
X 1 x — 7T X-\-Tt 1 X — 2 7T 1 X -J- 2tT 1 
1 . 1 J n 
— -l - pr~ d - , ^ — &C. 
-IS X + 7T 1 x — 2 or 1 r 
x+2i r 
3 T 
MDCCCLXXV. 
