ME. J. W. L. GLAISHEE ON THE THEOEY OF ELLIPTIC FUNCTIONS. 
497 
(Leyden, 1867), and deduced therefrom the resulting identities. The only formulae so 
obtained which appeared of interest were, in fact, those which are given in the present 
paper, viz. (18) to (23); but at the time I was not aware of their connexion with the 
theory of Elliptic Functions. It was only recently, after obtaining the values of 
sin am x &c. in (10) to (17), that I remarked that the resulting identities w r ere the same 
as those which I had previously deduced by the aid of Sir W. Thomson’s principle. 
It was shown by Cayley at the end of Sir W. Thomson’s paper that the identity (28) 
corresponds to 
Q(ui, H(m+K\ K)\ ..... (29) 
and it is singular that all the identities that follow from the method of this section thus 
appear to correspond either to elliptic or theta-function transformations. Speaking 
generally, the only evaluable integrals of the requisite form are derived from 
J « --w cos2 bxdx=^e~* and f «""cos bx dx — a*+b* 
(including as derivations the corresponding sine formulae), of which the former give rise 
to theta-function relations, and the latter to elliptic-function relations. 
§ 8. The integrals that produce the formulae (18) to (23), and the manner in which 
the latter are obtained from them, deserve some attention. Thus 
J* ~~ x dx~^ cosnx(e~ x —e~ 3x -\-e~ ix —&c.)dx 
1 3 5 
ra 2 +l 2 n 2 + 3 2 ‘ n 2 + 5 2 
it rnt 
=4 sech 
whereby (18) and (20) follow at once from (25) and (24). 
In a similar way we can show that 
? °° sin nx 
l o 
it .nit 
dx — ^ tanh — 
7 r e nn — 1 
4 e n7r + 1 ’ 
but the series obtained from the direct application of this integral would not converge : 
and in order to deduce (21) and (23) from (26) and (27), it is necessary to express the 
integral in the form 
and to make use of the formulae 
-Lcot|^ =sin 0+sin 20-}-sin 30+&c., 
\ cosec 0=sin tf+sin 3^ + sin 5^+&c. 
This renders the process not so satisfactory from a logical point of view ; but practi- 
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