464 
A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTIONS. 
[704 
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 simul 
taneously 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 analogous to each other; but on the one hand the course of the single theory 
would be only with difficulty perceptible in the greater complexity of the double 
theory ; and on the other hand we need the single theory as a guide for the course 
of the double theory. 
I accordingly stop to point out in a general manner the course of the single 
theory, and, in connexion with it but more briefly, that of the double theory ; and 
I then, in the Second and the Third Parts respectively, consider in detail the two 
theories separately ; first, that of the single functions, and then that of the double 
functions. The paragraphs of the Memoir are numbered consecutively. 
The definition adopted for the theta-functions differs somewhat from that which 
is ordinarily used. 
The earlier memoirs on the double theta-functions are the well-known ones :— 
Rosenhain, “Mémoire sur les fonctions de deux variables et à quatre périodes, qui 
sont les inverses des intégrales ultra-elliptiques de la première classe.” [1846.] Paris : 
Mém. Savans Étrang., t. xi. (1851), pp. 361—468. 
Gòpel, “ Theorise transcendentium Abelianarum primi ordinis adumbratio levis,” 
Creile, t. xxxv. (1847), pp. 277—312. 
My first paper—Cayley, “On the Double 0-Functions in connexion with a 16-nodal 
Surface,” Crelle-Borchardt, t. lxxxiii. (1877), pp. 210—219, [662]—was founded directly 
upon these, and was immediately followed by Dr Borchardt’s paper, 
Borchardt, “Ueber die Darstellung der Kummersche Flache vierter Ordnung mit 
sechzehn Knotenpunkten durch die Gópelsche biquadratische Relation zwischen vier 
Thetafunctionen mit zwei Variabeln,” Ditto, pp. 234—244. 
My other later papers, [663, 664, 665, 697, 703], are contained in the same Journal. 
FIRST PART.—INTRODUCTORY. 
Definition of the theta-functions. 
1. The ^)-tuple functions depend upon \p (p — 1) parameters which are the co 
efficients of a quadric function of p ultimately disappearing integers, upon p arguments, 
and upon 2p characters, each =0 or 1, Avhich form the characteristic of the 4^ functions; 
but it will be sufficient to write down the formulae in the case p = 2. 
As already mentioned, the adopted definition differs somewhat from that which 
is ordinarily used. I use, as will be seen, a quadric function J (a, h, bQm, n)- with