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XXL A Memoir on the Single and Double Theta-Functions. 
By A. Cayley, F.R.S., Sadlerian Professor of Pure Mathematics in the 
University of Cambridge. 
-Received November 14,—Read November 28, 1879. 
The Theta-Functions, although arising historically from the Elliptic Functions, may 
be considered as in order of simplicity preceding these, and connecting themselves 
directly with the exponential function (e r or) exp. x ; viz., they may be defined each 
of them as a sum of a series of exponentials, singly infinite in the case of the single 
functions, doubly infinite in the case of the double functions ; and so on. The number 
of the single functions is =4; and the quotients of these, or say three of them each 
divided by the fourth, are the elliptic functions sn, cn, dn ; the number of the double 
functions is (4~=) 16 ; and the quotients of these, or say fifteen of them each divided 
by the sixteenth, are the hyper-elliptic functions of two arguments depending on the 
square root of a sextic function : generally the number of the p-tuple theta-functions 
is =4' ; ; and the quotients of these, or say all but one of them each divided by the 
remaining function, are the Abelian functions of jo arguments depending on the 
irrational function y defined by the equation F(as, y) = 0 of a curve of deficiency p. 
If instead of connecting the ratios of the functions with a plane curve we consider the 
functions themselves as coordinates of a point in a (4 p —l)dimensional space, then we 
have the single functions as the four coordinates of a point on a quadri-quadric curve 
(one-fold locus) in ordinary space; and the double functions as the sixteen coordinates 
of a point on a quadri-quadric two-fold locus in 15-dimensional space, the deficiency 
of this two-fold locus being of course = 2. 
The investigations contained in the First Part of the present Memoir, although for 
simplicity of notation exhibited only in regard to the double functions are, in fact, 
applicable to the general case of the p-tuple functions ; but in the main the Memoir 
relates only to the single and double functions, and the title has been given to it 
accordingly. The investigations just referred to extend to the single functions ; and 
there is, it seems to me, an advantage in carrying on the two theories simultaneously 
up to and inclusive of the establishment of what I call the Product-theorem : this is 
a natural point of separation for the theories of the single and the double functions 
respectively. The ulterior developments of the two theories are indeed closely 
MDCCCLXXX. 5 Z 
