1879.] Prof. Cayley on Single and Double Theta- Functions. 397 



IV. " A Memoir on the Single and Double Theta-F unctions." 

 By A. Cayley, F.R.S., Sadlerian Professor of Pure Mathe- 

 matics in the University of Cambridge. Received Novem- 

 ber 14, 1879. 



(Abstract.) 



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- v 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. Tlie 

 number of the single functions is=I ; arid the quotients of these, or say 

 three of the functions, each divided by the fourth, are the elliptic func- 

 tions, sn, cn, dn : the number of the double functions is (4 2 =) 16 ; and 

 the quotients of these, or say fifteen of the functions, 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 p-tuple 

 theta-functions is =4? ; and the quotients of these, or say all but one of 

 the functions, each divided by the remaining function, are the Abelian 

 functions of jo arguments depending on the irrational function y 

 defined by the equation ¥(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 

 (4^—1) dimensional space, then we have the single functions as the 

 four coordinates of a point on a quadriquadric curve (one-fold locus) 

 in ordinary space : and the double functions as the sixteen coordinates 

 •of a point on a quadriquadiic (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 fnnctions, are in fact applicable to the general case of the 

 j>-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 establish- 

 ment 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 ac- 



