704] 
463 
704. 
A MEMOIR ON THE SINGLE AND DOUBLE THETA- 
FUNCTIONS. 
[From the Philosophical Transactions of the Royal Society of London, vol. 171, Part ill., 
(1880), pp. 897—1002. 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 x 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 2 =) 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 
^3-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 p arguments 
depending on the irrational function y defined by the equation F {x, 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 space of 
(4^—1) dimensions, 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