898 
PROFESSOR A. CAYLEY ON THE SINGLE 
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 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 :— 
Posenhain, “ Memoire sur les fonctions de deux variables et a quatre periodes, qui 
sont les inverses des integrates ultra-elliptiques de la premiere classed [ 1846.] Paris : 
‘ Mem. Savans Etrang.’ xi. (1851), pp. 361-468. 
Gopel, ‘ Theorise transcendentium Abelianarum priini ordinis adumbratio levis. 
‘ Crelle,’ xxxw (1847), pp. 277-312. 
My first paper— Cayley, “ On the Double d-Functions in connexion with a 
16-nodal Surface,” £ Crelle-Borchardt,’ lxxxiii. (1877), pp. 210-219—was founded 
directly upon these, and was immediately followed by Dr. Borchardt’s paper, 
Borchardt, “Ueber die Darstellung cler Kummersche Flitche vierter Ordnung mit 
sechzehn Knotenpunkte durcli die Gope/scheii Pelation zwischen vier Theta-functionen 
mit zwei Yariabeln.” Ditto, pp. 220-233. 
My other later papers are contained in the same Journal. 
FIRST PART.—INTRODUCTORY. 
Definition of the tlieta-functions. 
1. The p-tuple functions depend upon Jp(p+1) parameters which are the co¬ 
efficients of a quadric function of p ultimately disappearing integers, upon p argu¬ 
ments, and upon 2p characters, each =0 or 1, which form the characteristic of the 
P p functions ; but it will be sufficient to write down the formulae in the casey>=2. 
As already mentioned, the adopted definition differs somewhat from that which is 
ordinarily used. I use, as will be seen, a quadric function f(a, h, bfm, n)~ with even 
integer values of m, n, instead (a, h, bfrn, nf with even or odd values; and I write 
the other term \m(mu-\-nv) instead of mu-\-nv ; this comes to affecting the arguments 
u, v with a factor rri, so that the quarter periods (instead of being rri) are made to 
be =1. 
2. We write 
"j=^(a, h, bfm, n)~+^ni(mu-\-nv), 
