MR. J. W. L. G-LAISHER ON THE THEORY OF ELLIPTIC FUNCTIONS. 501 
+ cosech (#+/«/) + &c., tanh x — tanh ( x — fx) — tanh(#-f-| M ')+&c., and coth x — coth (x—fx) 
— coth (,r+ l a.) + &c., there is none for either 
or 
tanh ^4-tanh(^ — j«,) + tanh 
coth x-\- coth (x-[x)-\- coth (#+^)-|-&c. ; 
it is therefore interesting to inquire what are the corresponding formulae in these cases. 
If we write (21) in the form 
2 <tt ( it 2 Qttqc 2 2 4.7 TOC | 
cosech x + cosech (x -[x)-\- cosech (x-\-[x) + &c. = tanh — sin — -+- tanh — sin — - + &c. j, 
and reciprocate it by the third pair of formulae of § 9, we obtain the following result, 
tanh #-{-tanh (#— jO.)-|-tanh (#d-|H,)-b&c. 
2ir( 7T 2 . 2irx 2tt 2 . 4%x „ ) . 
= — ^cosech— sin — 4-cosech — sin — -f&c. >, (31) 
p-l V- v- v- P ) v 
which apparently ought to be the first of the two formulae sought ; but in point of fact 
this equation (as can be shown by actual calculation, see § 16) is not true. 
It seems natural to recur to the integral (30), viz. 
r 
mJ 
tanh 
dx 
cos nx^ m+1),r I - cosh# - ] 1 
1 Jo J 
2 . rnr 
■- + 7 T cosech - 77 , 
n 1 v. 7 
from which, since the first two terms of the right-hand member vanish when n is 
even, we have 
7»(m+l V 
J —mix 
, . 2n%x p m 2 
tanh x sin = — — -\-ir cosech — ; 
ju. me 1 jw, 
whence ultimately, since \tt — ^ 0 =sin 0-|--|sin 2^+^ sin 30-f-&c., 
tanh x + tanh (x—fx)-\- tanh (x -j- (x) -f &c. 
=— — 1 +— < cosech — sin — - 1 - cosech — sin Ji:i::: -|-&c. . . . 
**■ P l V- P f* P 
but this result is not true either, and for the following reason : — Let 
^x=Qx—<p (x—fx)—(p (x-{-[x) . . . +<p (x— n[x)+<p (x+np), 
%x=<px+<p (x—(x) + q> (x+ix) . . . +<p (x— n(x)+<p (x+np) 
2tt 2 . 4 irx 
and 
(n infinite), and suppose <px is an uneven function of x which = 1 , when x— 00 . 
Then 
4 /(x-\-fx)=—fyx+<p(x—nfx)+<p (x-\-(7i-\-\) p) 
= —-tyx, 
( 32 ) 
