50 
[811 
811. 
ON THE LINEAR TRANSFORMATION OF THE THETA FUNCTIONS. 
[From the Messenger of Mathematics, vol. xiii. (1884), pp. 54—60.] 
The functions referred to are the single Theta Functions ; these may be defined 
as doubly infinite products, as was in fact done in my “ Mémoire sur les fonctions 
doublement périodiques,” Liouv. t. x. (1845), pp. 385—420, [25] ; and it is interesting to 
consider from this point of view the theory of their linear transformation : this I propose 
to do in the present paper, adopting throughout the notation of Smith’s* “Memoir on 
the Theta and Omega Functions.” 
The periods K, iK' are, in general, imaginary quantities 
K =A + Bi, 
iK' = C+Di, 
where AD — BG is positive ; writing then &> = , and q = e { ™, also for shortness 
(gl) =2g*Iir (1 -g m ) 3 , 
where gl denotes e^™, the expression of the odd theta-function ^ (x, w) as a doubly 
infinite product is 
S'! {x, co) = (gl) ¿dill ^1 + 
x 
mir + nwir. 
where (m, n) have any positive or negative integer values (the combination m— 0, n = 0 
excluded) from m = — ¡i to g, and n = — v to v, g and v being each ultimately infinite 
but so that g is infinite in comparison with v; this condition in regard to the limits 
is indicated by /¿/v = oo; and similarly v/g = oo would indicate that v was infinite in 
comparison with /x. 
t* Smith’s Collected Mathematical Papers, vol. n., pp. 415—621.]