[ f 83 ] 
XVII. Memoir on the Theta-Functions, particularly those of two variables. 
By A. R. Forsyth, B.A ., Fellow of Trinity College , Cambridge. 
Communicated by Professor A. Cayley, LL.D ., F.R.S. 
Received December 9,—Read December 22, 1881. 
The following paper is divided into four sections. Section I. deals with what may be 
called Rosenhain’s theory ; under the guidance of Professor H. J. S. Smith’s paper 
on the single theta-functions (in vol. i. of London Math. Soc. Proc.), there is investi¬ 
gated a general theorem for the product of four double theta-functions with different 
characteristics and variables, the definition being 
cjp J P jx, y 1 = % % (_ 
L XU) V j J Ml =—00 W =-00 
the product being equal to the sum of 16 similar products; and the equation is shown 
to include 4096 particular cases. Quadratic relations are established between the 
functions; and the 15 quotients of all of them but one by that one are expressed in 
terms of two new variables x 1} x 2 , the connexion between x 2 , x 2 and the original 
variables x, y being 
x 
-i 
H 
^A+Bs 
y/Z 
Bs , , p 
T dz+\ 
v/Z dz+ \ 
^A + B z 
dz 
•v/Z 
y/Z 
dz 
where 
Z=z(l —z)(l — Kl h) (1 — k 2 2 z) ( 1 — k 2 z) 
and A, B, A', B', * lf k 2 , k s are perfectly determinate constants. The quadruple 
periodicity of the functions is investigated at the beginning of the section, and at 
the end definite-integral expressions for the periods are obtained. 
Section II. gives the expansions of all the functions 
(i) in trigonometrical series ; 
(ii) in ascending powers of x and y. 
To obtain the latter, use is made of a theorem there proved :— 
5 H 
MDCCCLXXXII. 
