MR. J. W. L. GLAISHER ON THE THEORY OE ELLIPTIC EUNCTIONS. 
511 
which was required in the verification, is best obtained by deducing it from the well- 
known theorem 
cos a? i cos 2a? t cos 3a? t t cosh (/3x — /3a?) 1 
l 2 + /3 2 + 2 * + fi* + 3 2 ^ C ’ = 2/3 sinh fir 2/8*’ 
from which, by writing ^/3 for 0 and 2x for x , dividing the equation so obtained by 4, 
and subtracting it from (47), we find 
cos a? , cos 3a? p _ n (cosh (/Sar — /3a?) 1 cosh(^/37r— /3a?)) 
l 2 + /3 2 ' 3 9 + ^ 2 + &C, — 2^ ( sinh /S* 2 sinhphr j 
7r j cosh(/3?r— /3a?) — cosh /3a?) cosh ^/Stt) 
2/3 1 sinh /37r j 
?r sinh (^/37r— /3a) 
4/3 cosh 
It is to be noticed that (46) is only true if x lies between 0 and /a. This may be 
regarded as a consequence of the fact that (47) only holds good when x is positive 
and less than 2-jt ; but the necessity for the condition is also evident from the process of 
verification by ordinary algebra. Thus the expression in (46) 
= 2-je x —e 3 
2 cosh a? 
l+e'* 
2 cosh 3a? ) 
2e~ 
1+e 
^-2(e x + e~ x )(e-» - e~^+e- 3 » - &c.) + 2 (e 3x + e~ 3x )(e-^ - + e~ 9 * -...)- &c. 
e x + e~ x 1+e 2 ^"/ 1 ) 
OpX+IL 
& c ° 
= sech x — sech (x—p) — sech (x-+- p) -f- &c. , 
wherein we see that to justify the summations of e~ x —e~ 3x -\-&c , and e x ~* — e 3(J: " M) +&e. 
as ordinary geometrical progressions we must suppose x to be positive and less than [x. 
Also since sech x— sech (x— p) — sech(.r+ j u»)-f-&c. is periodic, while the expression in 
(46) is not so, we see that the equality will not hold good beyond these limits. 
I have worked out the corresponding proofs of the other five identities (19) to (23) 
in the same way, but none of them call for any special remark. The process is not in 
all cases exactly similar, as, ex. gr., in deducing (19) from 
sinh/3i— (3 jS a +l^/3 s +2 e Kc ’> 
sin a? 3 sin 3a? 5 sin 5a? p w cosh (^/3w— /3a?) 
/S 2 + 1 2 ‘ j3 2 + 3 2 ‘ /3 2 + 5 2 C ‘ 4 cosh^/3flr ’ 
^Sf;+5ffif, +&c -=^( sin *+i fdn3 *+I sin5ji + &c -) 
/x ( cosh (p — 2 a) cosh2(ju. — 2a?) „ ) 
5r | cosh/u. cosh 2ju. ' C ’j ’ 
we find 
