502 MR. J. W. L. GLAISHER ON THE THEORY OF ELLIPTIC FUNCTIONS. 
so that -tyx is periodic ; but 
X 0* +^)=X^-P ( x ~ np)+Q (#+(» + !) /*) 
so that yjx is not periodic. Therefore we have no right to assume that between the 
limits 0 and \x of x 
tanh a’+tanh (x— /a) -|- tanh (x-\-p)-{-&c. 
can be expressed in the form 
. . 2 nx , . . 47 xx . 6nx 
A l sm -f-A 2 sin— -+A 3 sm + &c., 
the true form being 
-D • ™ I -D * 2%X , T> • 3,r X I D 
fc>, sm ■ — j-J3 2 sm — 4-.B3 sm — -f-&c. 
f p 
We may, however, assume that between the limits 0 and of x 
tanh x-\- tanh (x — ,«/)-)- tanh (#-|-^)-f-&c.=A 1 sin ^-+A 2 sin -(-&c. ; 
and then 
\ tanh x -T tanh ( x — p) tanh (x +//,) -|-&c. j- sin - — dx 
+&c.| 
tanh x sin —— fa 
^(2m+l) ; 
2, r • 2mrx 7 
tanh x sm dx 
r a, 2rntx^ imJrl ^ C 2 
= — o — cos —4 ; - -rs: 
L 2n% p _|o Jq e + 1 
2mrx 
sin dx 
V- 
= (-) n+1 cosech—. 
^ > 2nn 1 2 ju. 
We thus find that between the limits 0 and of x (and therefore also between 
the limits — ^ and %/& of x) 
4tCX 
tanh tanh (x — ^)+tanh {x-\-^)-\-8c c.=“ ‘ sin ^ sin. -h &c. j- 
, , 5T 2 . 27ra? , 2?r 2 . 4tx 
cosech — sm — ■ + cosech — sm — -4-&c. 
2x 2i t( 7 r 2 . 2-7TX . , 27 t 2 . 4%x ) 
=- — — -{cosech — sm \- cosech — sm — +&c. >, 
v- /M v- v- v- v- ) 
. (33) 
the terms on the left-hand side being uneven in number, and such that for every term 
tanh (x—np) there is also a term tanh (x-\-nyj). 
If we write x+p for x in this formula (33) we increase the left-hand side by 
9 
tanh co -(-tanh co , that is by 2, while the right-hand side is increased by - . that is 
