704] A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTIONS. 
479 
B. A 
A. B 
D.C 
C. D 
I).A 
A . D 
B. G 
G.B 
^ Q U + u' Sr U — u' = 
0 ” 0 ” “ 
^^= 
^1 7} ^1 " 
. 0 .1 
^ J u + u' ^ u — v! — 
a 1 
^ 0 yy ^ j ” 
PP' + QQ\ (second product-set) 
PQ' + QP', 
iPP' - iQQ\ 
iPQ' — iQP'; 
P,P,' + Q,Q f ', (third product-set) 
iPM-iQjp;, 
iP,p; + 
p,q;+ qa- 
33. Here, and subsequently, we have 
®0’ ©J, ®J, ®î<2») = *, Y, X,, Y, 
„ „ „ „(2u')=x\ r, x;, y; 
» » » 3) (0) = ot , /3, cl /} (3 t 
® 2 0 , ®l, ©|(2tO =p, Q, P /} Q„ 
„ „ „ ,,(2u') = P', Q', Pf, q;, 
33 33 33 33 (0) p, q, p,, q,, 
viz. we use also a, (3, a,, ¡3and p, q, p,, q, to denote the zero-functions; ¡3, is = 0, 
but we use /3/ to denote the zero-value of Y,. 
34. In order to obtain the foregoing relations, it is necessary to observe that 
© 
a + 
7 
by which the upper character is always reduced to 0, 1, \ or f; and that, for re 
ducing the lower character, we have 
„ 0 „ 0 „ 1 „1 
© =©;© = — © 
y+2 7 7+2 7 
(Hi 2 
© 2 „ = ¿0 2 
7+2 7 
© 2 = - ¿0 2 , © 2 n = ¿0 2 ; 
7+2 7 7—2 7 
by means of which the lower character is always reduced to 0 or 1: in all these 
formulae the argument is arbitrary, and it is thus = 2u, or 2u' as the case requires. 
The formulae are obtained without difficulty directly from the definition of the 
functions ©.