510 MR. J. W. L. GLAISHER ON THE THEORY OF ELLIPTIC FUNCTIONS, 
7 sech 77 - cos 32=^2 
4 2 ft Tt z 
5n 
7 sech — cos 5z=^_ 
cos 3 z 
3 cos 3 z 
^ + 3 2_ 
W^' 
7* 
1t i 
cos 5z 
3 cos 5 z 
1 — 4- 5 2 ” 
,r 2 + 5 
I oa 
1 »0 
+ 
tn 
5 cos 3z 0 
-4~ To 2~? &C. 
5 V „„ 
5 cos 5z 
Vll 1 
rJ 'L . 5-2 
— &c. ! 
whence 
2tt ( . Ti- 2 ■ 3 tt 2 0 . 5tt 2 _ D 
— -jsech^ cos 2 + sech — cos 02 -f- sech cos 02 + &c. 
+ &c. 
8ju. ( cos . 
cos 3 z 
cos 5 : 
“f - T72 
;+P 
3 2 
■ 5 2 
( 3 cos z 
3 cos 3 z 3 cos 5 z 
te* + 
* 
2V +s « + ^ + 5’ 
7T Z 7T Z 
+ &C. j_ 
+ ....• 
^Jsinh (ijx— a?) sinh 3 (^//.— a?) sinh 5 (|)x — a?) 
I cosh cosh -|jx cosh /x 
2 cosh a?\ / 2 cosh 3a? 
&c.| 
1 +e^ 
) +&c.| 
4 cosh a? 4 cosh 3a? „ 
" l +e ^ + 1 +e^ ~ &c -’ 
which, as shown in § 4, 
= sech x — sech (x—p) — sech (x + + &c. 
We thus in the course of the proof obtain another form for sech x — sech (x—p) 
'-sech(ar+^)H-&c., viz. 
o (sinh (i/*-— a?) sinh3(i^-a?) sinh 5 (ift-a?) ) . 
cosh 7 jx ~ coshfp, ■+" coshf/x — «c.| V 4U ! 
whence, in addition to the forms for cos am u in § 14, we have 
it fsinhfiv— z) sinh 3 (iv— z) n ) 
cosam«=jg,|- oshi> — coshfy + &<=•{• 
This method of proof is not so interesting as that of § 11, both because the formulae 
required cannot be obtained in so elementary a manner, and also because the identities 
(18) to (23) are not so directly verified, as their right-hand members are shown to be 
equal to expressions such as (46), which themselves need some transformation before 
they assume the desired forms. The formula 
cos a? cos 3a? 
it sinh (|/3 tt— / 3a?) 
