550 
A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTIONS. [704 
and we then have further 
that is, 
C 4 C 8 ^o^i- — fàg + C 3 Cig^'7^"ii, 
= CqCy/I'Z + C-jfiizjJW 
whence equating the two values of Sv^is 2 we have the required quartic equation in 
X, y, z, w. 
123. But the reduction is effected more simply if instead of the c’s we introduce 
the rectangular coefficients a, b, c, &c. We then have 
M — (c" 2 — b' 2 ), A = — a"c, C = a'b, 
B = - b'c' - b"c", =bc, D = b'b" + c'c', = a'a"; 
and the equations become 
(c" 2 — b' 2 ) \ 2 = — a"cx 2 + bey 2 + abz 2 — ci'a"w 2 , 
(c" 2 — b' r ) b- 12 3 = a'bx 2 — a'a"y 2 — a'cz 2 + bew 2 , 
V b'c'^f 0^12 = Va xz + V— b"c'yw, 
so that the elimination gives 
b'c" (— a"cx 2 + bey 2 + a'bz 2 — a! a"w 2 ) (a'bx 2 — a'a" y 2 — a'cz 2 + bew 2 ) 
= (c" 2 — b' 2 ) 2 [ax 2 z 2 — b"c'y 2 w 2 + 2 V— ab"c'xyzw\, 
viz. this is 
— a'a"bb'cc" (x* + y 4 + z* + w 4 ) 
+ a'b'cc" (a" 2 + b 2 ) (x 2 y 2 + z 2 w 2 ) 
+ {b'c" (d 2 b 2 + a" 2 c 2 ) — a (b' 2 — c" 2 ) 2 ) x 2 z 2 
+ [b'c" (a' 2 a" 2 + b 2 c 2 ) + b"c' (b' 2 - c" 2 ) 2 } y 2 w 2 
— a"bb'c" (a 2 -f c 2 ) (x 2 w 2 + y 2 z 2 ) 
— 2 (b' 2 — c" 2 ) 2 V— ab"c'xyzw = 0. 
124. In this equation the coefficients of x 2 z 2 and y 2 w 2 are each = da"bc (b' 2 + c" 2 ) r 
as at once appears from the identities 
a'b. b' — c". a"c = a(b' 2 — c" 2 ), 
a'b. c" — b'. a"c = (b' 2 - c" 2 ), 
a'a" .b'-c" .be = - b" (b' 2 - c" 2 ), 
\a'a".c - b'.be = c' (b' 2 - c" 2 ), 
by multiplying together in each pair the left-hand and the right-hand sides respec 
tively. Substituting and dividing by — a'a"bb'cc", we have 
x 4 Ay 4 + z 4 + w 4 
a" 2 + b 2 
a"b 
(x 2 y 2 + z 2 w 2 ) — 
b' 2 + c" 2 
b'c" 
(x 2 z 2 + y 2 w 2 ) + 
a 2 + c 2 
a'c 
(x 2 w 2 + y 2 z 2 )