64 Mr Glaisher, On expressions for the [May 7, 



r being arbitrary subject only to the condition that it must be 

 greater than unity in the first formula, and greater than zero in 

 the second. Putting r = 2, we thus obtain for the theta function 

 the expression^ 



&(x + K) = —e^ KK- \ e i" \ — coBxt 



vot, 



cos xt — eK' cos {x-^K)t 



+ 



1 — 2e" Z^ cos Kt + e~K^ 



cos xt — e~'K^ cos {x + K) {[ , 



1 — 2e K' cos Kt + e k' 



§ 6. A third method of obtaining the value of © {x) is by 

 means of the integral 



f 



Jo 



cos Ct ,^ , TT „^ ... 



dt=^ - e-"" (4), 



we thus have 



2a' 2(a-6? 2(a + 6? , \ 



+ . ,x4 . .2 + . , \ ; 4 . ,. + &c. cos tdt, 



and it is found that 



4i 4'^4i/ N4T^4i/ 1 \4~ O0<j. 



X +a X +ia — TT) X + {a + tt) 



"" »V2 1*^^'^' ^^ + */>(*'- «)| ' 



where (fcTaj al - ^mh ^2 - sin (a:V2 + 2a) 



leading to the formula 



@(x)=a/ - . I {(f> (pit, u)+<f) {at, - u)] cos (f) dt, 



1 The three formulae in this section were given without proof in the Messenger 

 of Mathematics, t. v. p. 173 (March 1876) : but there are several misprints, and in 

 the expression for G^, the x that occurs in the exponentials should be replaced by 



x-k: 



