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A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTIONS. 
555 
Again, c 4 c 8 k 7 k n , = \Zacf. bde y/bcf. ade v 7 acclef v 7 bcdef is a fourth root of a product of 
32 factors, which are in fact 16 factors twice repeated, and in the 16 factors, ab does 
not occur, cf and de occur each twice, and the other 12 factors each once: we thus 
have 
c 4 cjc 7 k n = li J (cf) 2 (de) 2 -v- (ab)' 2 , 
and the relation to be verified assumes the form 
fc.deV 1 -T- (cf) 2 (de) 2 + V(cf ) 2 (de) 2 — 0, 
which, taking fc .de = — \/ (cf) 4 (de) 4 , is right. And so for the other equations. It will 
be observed that, in the equation de .fc . c 3 c u c 8 c n + c 4 c 8 k 7 k n = 0, and in the other equations 
upon which the verifications depend, there is no ambiguity of sign: the signs of the 
radicals have to be determined consistently with all the equations which connect the 
c’s and the k’s: that this is possible appears evident a priori, but the actual verification 
presents some difficulty. I do not here enter further into the question. 
Further results of the product-theorem, the u ± u formulae. 
132. Recurring now to the equations in u + u, u — u', by putting therein u! — 0, 
we can express X, Y, Z, W in terms of four of the squared functions of u, and by 
putting u = 0 we can express X', Y', Z', W in terms of four of the squared functions 
of u'; and, substituting in the original equations, we have the products 
^ ( ) u + u!. ^ ( ) u — u! 
in terms of the squared functions of u and u'. 
Selecting as in a former investigation the functions 4, 7, 8, 11, which were called 
x, y, z, w, it is more convenient to use single letters to represent the squared functions. 
I write 
'à- (u + u') .^f(u — u') 
X 2 u 
X-ii 
A 2 0 
4 
4 = P, 
II 
II 
4 = p 0 (=c 4 2 ), 
7 
7 = Q, 
7 = q, 
*<r 
II 
7 = 0, 
8 
8 = R, 
8 = r, 
8 = /, 
8 = r 0 (=c 8 2 ), 
11 
11 = S, 
11 = s, 
11 = s', 
11 = 0. 
Then 
X 
F = X' — Y' + Z' — W, 
Q = W'-Z' + Y’-X', 
R = X' + Y'-Z' - If', 
8 = W'+Z' - Y'-X', 
Hence 
X Y Z W 
p= a — ft + y — 8, 
q = 8 — y + ft —a, 
r — a + ft— y — 8, 
s = 8 + y- ft — a, 
X' Y' Z' W' 
p = a — ft + y — 8, 
(/ = 8 — y + ft — 0L, 
r = a + ft — y — S, 
s' = 8 + y — ft — a. 
a(p + r) — 8 (q + s) = 2 (a 3 — 8 2 ) X, 
8 „ —a „ =2 „ W, 
- ft(p-r) + y(q-s) = 2(ft 2 -y 2 ) Y, 
— y ,, +ft » =2 „ Z, 
a (p' + r') -8(q' + s') = 2 (a 2 - 8 2 ) X', 
8 „ — a „ =2 „ W, 
-ft (p'~ r') + y(q'- s') = 2 (ft 2 - y 2 ) T, 
— 7 » + ft » =2 „ Z'. 
70—2