704] 
A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTIONS. 
471 
In the second case, writing /x +1, f +1 for ¡x, f, the new values of /x, f will 
be both even, and we have the like expression with only the characters \ (a + a), 
h, (a — a!) each increased by 1; so in the third case we obtain the like expression 
with only the characters ^ (/3 + /3'), ^ (/3 — ¡3') each increased by 1; and in the fourth 
case the like expression with the four upper characters each increased by 1. The 
product of the two theta-functions is thus equal to the sum of the four products, 
according to the theorem. 
Resume of the ulterior theory of the single functions. 
15. For the single theta-functions the Product-theorem comprises 16 equations, 
and for the double theta-functions, 256 equations: these systems will be given in 
full in the sequel. But attending at present to the single functions, I write down 
here the first four of the 16 equations, viz. these are 
0.0 
Ko) 
(u + u').^f ( 
q)o-o= 
XX'+YY', 
1.0 
rH o 
à1 
„ ^ 
1 
0 
YX' + XY', 
0.1 
a 0 
l 
» * 
0 
1 
XX' — YY\ 
1.1 
1 
„ * 
1 
1 
- YX' + XY'- 
where X, Y denote 0 ^ (2u), 0 (J^j (2u) respectively, and X', Y' the same functions 
of 2u! respectively. In the other equations we have on the left-hand the product of 
different theta-functions of u + v!, u — u respectively, and on the right-hand expressions 
involving other functions, X 1} Y lt Xf Yf &c., of 2u and 2u respectively. 
16. By writing u' = 0, we have on the left-hand, squares or products of theta- 
functions of u, and on the right-hand expressions containing functions of 2u: in 
particular, the above equations show that the squares of the four theta-functions are 
equal to linear functions of X, Y; that is, there exist between the squared functions 
two linear relations: or again, introducing a variable argument x, the four squared 
functions may be taken to be proportional to linear functions 
31 (a — x), 33 (b - x), (5 (c — x), 2) (d — x), 
where 21, 33, (£, 2), a, b, c, d, are constants. This suggests a new notation for the 
four functions, viz. we write 
Ko)«- KS) W ’ 
* (J) OX 
Ki) (,i) 
= Au, Bu, 
Gu, 
Du ; 
and the result just mentioned then is 
A 2 u : Bhi : 
Chi : 
D 2 u 
II 
1 
1 
: (£ (c — x) : 
3) (d — x),