AND DOUBLE THETA-FUNCTIONS. 
909 
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\ and writing for shortness 
fa— f a — x.a — y, 
f ab — r _ y {f ' a—x.b—xf—x.c—y.d—y.e—yf f a—y.b—yf—y.c—x.d — x.e—x}, 
(observe that in the symbols fab it is always f 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 abfcde ) ; then the 16 functions are proportional to properly determined 
constant multiples of 
fa, fb, fc, fd, f e, ff fab, fac, fad, fae, fbc, fbd, fbe, fed, fee, fde, 
and this suggests that the functions shall be represented by the single and double letter 
notation A(w, 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, BC, BD, BE, CD, CE, DE 
proportional to determinate constant multiples of the before-mentioned functions 
fa, . . f ab, ... of x and y. 
24. It is interesting to notice why in the expressions for f ah, &c., the sign con¬ 
necting the two radicals is -f- ; 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 f a, 
&c., unaltered, while it changes fab, &c., into — fab, &e. ; 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—vJ U+vJ It—vJ 
A . B - B . A, 
