514 
ME. J. W. L. GLAISHEE ON THE THEOET OF ELLIPTIC FUNCTIONS. 
(by use of the formula sin 3-f-| sin 25+^ sin 30 + &C — -|0) 
h 2 fc- 
sinh 2/x. 
lx ^ sinh Q«,-2a?) 2 sinh2 (^— 2a?) „ _ 
— I— ^ ^ sinh r 
and therefore 
2a? 
2n /sin 2* sin 4^ ~ 
ft 1 fxT ysmhv sinh 2v ' ^ C ‘ 
: 1 _ 2e- 21 + 2e" 4 *- &c. + 2 ( ^If 
sinh (ft — 2a?) ^ sinli 2(p,— S 
sinh ft ' sinh 2ft 
a?) 
■&c. 
e^—i 
-f- &c 
•) 
= l__ 3 l- 1 - l -2{(^-e- 2 ^ _2 "+e- 4 ' 1 +&c.)-(e 4I -e- 4l )(^ + e- 8 ' 1 +&c.) + &c \ 
( g2(2:- M ) e -2(i+/x) \ 
= tanh .z + 2| 1 + ea(I _ M j — 1+e - 2 ( *Uo + &c.| 
=tanh #-f-tanh (#— ^H-tanh (a?+jM/)+&c., 
which is the true formula. 
In the same way, since 
2tt 
2 - 2 - 
coth * sin 2z=— <- + — ^~ 2 +&c. >sin 2z, 
!« + i 2*+4 i 
we find that 
2w _ , . \ 2?r , _| „sinh (ft — 2a?) , „ sinh 2 (a— 2a?) . n 
-(coth. sin 2z+coth 2» sin 4* + &a)=--+l + 2 s i nh(t + 2 sinh 2,* + &c ' 
= — — 4-coth #+coth(.z— ^) + coth(a:+^)+&c., 
which is correct, and agrees with (36). 
It is of course easy to assure one’s self that (48) cannot be true ; for, taking for 
simplicity, and differentiating with regard to x or z , 
4 , 4 4 „ 0 ( cos 2a? 2 cos 4a? 3 cos 6a? ) 
(eZ + <? -x)2 + + e -(*-*>)2 + (e*+ir + C -<z+ir))a+ &C * — 0| e ir_ e -n + e 2 ff _ e -2 ff + < .3^_ e -3*r + &C *| »’ 
and it is evident that if we take x>\tt and < f 7 r we should have a positive quantity 
equated to a negative quantity. 
I thought it of interest to actually verify numerically the truth of the formulae (33) 
and (36) in one or two cases. Working with seven-figure logarithms, and taking ^ = 2, 
x=%, I found that each side of (33) was =0545188, and for 2, that each side 
was =0-282281 ; while for x=%, ^=2 each side of (36) was =2*07112, and for x=%, 
fjj=2 each side was =4*04247 ; placing beyond doubt the correctness of (33) and (36). 
It is a characteristic property of the identities noticed in this paper that in all cases 
the series on both sides are convergent whatever may be the values of x and For 
the actual calculation of the elliptic functions the formulse (10) to (17) would be 
preferable to (1) to (8) if the angle of the modulus was very near to 90°, so that q was 
