494 ME. J. W. L. GLAISHEE ON THE THEOEY OE ELLIPTIC EUNCTIONS. 
The comparison of (4) and (13) gives 
cosech (x—^ + cosech (^+^) + cosec h (%— if) + cosech +&c. 
. . 2nx A . 4%x 
4 sin 4 sm 
*(. 7TX ft ft ) 
=;]“ tan 7+^ 1 ? — + &c - 
* e* + 1 e'+l > 
which, on replacing x by x-\-^(Jj, becomes 
, 1 OliX 1 OiU -V 
cosech x -\- cosech (x— p )-\- cosech (^d-^) + &c.=-<cot— — ■ - 2 - 2 -— ~ ^ 2 — — — See. >. 
^ ^ e^ + 1 e^+l ) 
From (5) and (14) we deduce 
. . 2nx . . 4irx 
4 sin 4 sm 
. irx . . 3nx 
4 sin — 4 sin — 
coth#— coth (x— /a)— coth (#+^)+&c.=--jcosec — — — -f 3t2 ^- -{-&c. 1. (22) 
^ ^ e ~ — l e T-i > 
The comparison of (6) and (15) gives 
—cosech (x — cosech + cosech (^— 2 ) ~~& c - 
■n ( 
=-<sec — — - 
ttx . 3 irx 
4 cos — 4 cos 
A . irx . . 3irx 
4 sin — 4sin- 
s^+l e* +1 J 
which, on replacing x by becomes 
cosech#— cosech {x —^)— cosech (^+ i «/)+&c.=-< cosec — — &c. >. 
^ ^ «* + l e^ + 1 J 
The comparison of the forms for ^ ^ , (7) and (16), merely gives an equation which, 
on replacement of x by x-\ --^a, is identical with that resulting from A am u, viz. (20), 
while the forms of cot am u, (8) and (17), lead at once to (21). 
In the expressions on the left-hand side of (19) and (22) the number of terms included 
must be uneven. 
It is proper to remark that the formulae for <px—<f>(x—[^) — <p(#+|W')+&c. can be 
readily deduced from those for <p#+<p(#— p) -f <p(#+/a) +&c. ; thus (18) is a consequence 
of (20) and (23) of (21). For ex.gr. in (20) write 2 ^ for /a, and we have 
„ irx 2irx 
2 cos — 2 cos 
sech x + sech (x— 2^) + sech (#4-2fA)-|-&c.=^ jl-f Jr+&c. > 
^ t cosh — cosh — ' 
2/a /a 
