486 
A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTIONS. [704 
where observe that, in taking the difference, the right-hand side becomes divisible by 
a 2 — /3-, and therefore in the final result we have on the left-hand side the simple 
factor cl- — (3 2 instead of (a 2 — /3 2 ) 2 . 
Similarly 
(a 2 — /3 2 ) YX' = a/3 (ChiChi + D 2 uD 2 u') — a-D-uG-u’ — /3 2 ChiD 2 u', 
„ XY'=a/3 „ - /3 2 „ -a 2 „ 
and thence 
(a 2 - /3 2 ) 2 ( YX' + X Y') = 2a/3 (ChiChi' + DhiDht) - (a 2 + /3-) (GhiDht' + DhtChi), 
(a 2 -/3 2 ) (- YX' + XT) = DhtOhl- GhiJDhi. 
48. Hence recollecting that 
A 2 0 = a 2 + /3 2 , 
IPO = 2a/3, 
C 2 0 = a 2 - /3 2 , 
the original equations become 
C 4 0. A (u + u') A (u — u') = A 2 0 (ChiChi + D 2 uDhi) — B 2 0 (GhtDhi' + DhiChrf), 
CH). B (u + u') B (u — u r ) — IPO (G 2 uG 2 u' + D 2 uD 2 u') — A 2 0 (ChiDhtf + D 2 uG 2 u'), 
G 2 0 . C (u + u') G (u — u') = G 2 uG 2 u' — DhiDhi', 
C 2 0. D (u + u') D (u — v!) = D 2 uG 2 u' — G 2 uD 2 u'. 
49. It will be observed that the four products A (u + v!)A (u — u'), &c., are each 
of them expressed in terms of G 2 u, D 2 u, C 2 u', Dhi. Since each of the squared functions 
A 2 u, B 2 u, G 2 u, D 2 u is a linear function of any two of them, and the like as regards 
A 2 u, Bhi, Chi, D 2 u', it is clear that in each equation we can on the right-hand 
side introduce any two at pleasure of the squared functions of u, and any two at 
pleasure of the squared functions of u'. But all the forms so obtained are of course 
identical if, taking x the same function of u that x is of u, we introduce on the 
right-hand side x, x instead of u, v!; and the values of A (u + u'). A (u — u'), &c., 
are found to be equal to multiples of V, V 1} V 2 , V 3 , where 
V = x — x', V x = 
1, 
X + X, 
iAji/j 
, v 2 = 
1, 
x + x', 
xx' 
, v 3 = 
1, 
x + x', 
xx' 
1, 
ah- d, 
ad 
1, 
b + d, 
bd 
1, 
c + d, 
cd 
1, 
b + c, 
be 
1, 
c + a, 
ca 
1, 
a + b, 
ab 
50. In fact, from the equations 
A 2 u = 21 {a — x), A 2 u' = 21 (a — x), 
we have 
V = a K ~ GhiBhi ), 
= (A-hiI)hi — BhiAhi'), 
= b(I15 (C~ u A 2 u — A 2 uC 2 u), 
— g^X) (B 2u D 2 v! — DhiBhi'), 
= (AhiBhi' — BhiAhi), 
= (GhiDhl — D 2 uG 2 u),