474 
A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTIONS. [704 
Resume of the ulterior theory of the double functions. 
23. The ulterior theory of the double functions is intended to be carried out 
on the like plan. As regards these, it is to be observed here that we have not only 
the 16 equations leading to linear relations between the squared functions, but that 
the remaining 240 equations lead also to linear relations between binary products of 
different functions. We have thus between the 16 functions a system of quadric 
relations, which in fact determine the ratios of the 16 functions in terms of two 
variable parameters x, y. (The 16 functions are thus the coordinates of a point on 
a quadri-quadric two-fold locus in 15-dimensional space.) The forms depend upon six 
constants, a, b, c, d, e, f: writing for shortness 
fa— f a — x. a — y, 
fab = [fa — x.b — x ./— x.c — y.d—y.e — y + fa — y.b — y .f—y. c — x .d — x.e — x], 
x-y 
(observe that in the symbols fab it is always / that accompanies the two expressed 
letters a, b—or, what is the same thing, the duad ab is really an abbreviation for 
the double triad abf.cde): then the 16 functions are proportional to properly determined 
constant multiples of 
fa, fb, fc, fd, fe, ff, fab, fac, fad, fae, fbc, fbd, fbe, fed, fee, fde: 
and this suggests that the functions should be represented by the single and double 
letter notation A (u, v),..., AB(u, v),...; viz. if for shortness the arguments are omitted, 
then we have 
A, B, C, D, E, F, AB, AC, AD, AE, BG, BD, BE, CD, GE, DE, 
proportional to determinate constant multiples of the before-mentioned functions 
fa,..., fab,..., of x and y. 
24. It is interesting to notice why in the expressions for f ab, &c., the sign 
connecting the two radicals is +; the effect of the interchange of x, y is, in fact, to 
change (u, v) into (— u, — v); consequently to change the sign of the odd functions, 
and to leave unaltered those of the even functions: the interchange does in fact leave 
fa, Szc., unaltered, while it changes fab, &c., into — fab, &c.; and thus, since only 
the ratios are attended to, there is a change of sign as there should be. 
25. The equations of the product-theorem lead to expressions for 
u+u' u-v! u+u! u-u' 
A.B - B .A, 
where the arguments, written above, are used to denote the two arguments, viz. u + u' 
to denote (u + u', v + v') and u — u' to denote (u — v!, v — v') ; and where the letters 
A, B denote each or either of them a single or double letter. These expressions