AND DOUBLE THETA-FUNCTIONS. 
985 
b'c"( — a"cx~ + bey 2 + a -^ ~ a'a"iv 2 ) 
X ( ctbx-—a a"if — a"cz 2 + bcw~) — (c"° — b'~) 2 .( ctxh~ — b"c'yhv : ab"c'xyzw }, 
viz.: this is 
— a'a'bb'cc "(x*+y*+z*+w±) 
-\-a'b'cc" b~)(x 2 y z -{-z 2 w 2 ) 
+ {b'c"(a' 2 b 2 +a" 2 c 2 ) - a(b'- 2 -c"' 2 )*}x V 
+ {&'c"(flV 2 +^) _] \-b"c'(b' 2 -c"Y}fw 2 
-a''bb'c"(a 2 +c 2 ){x z w 2 +yh 2 ) 
— 2 (b '~— c"*)W-ab"c'xyzw= 0. 
124. In this equation the coefficients of x~z~ and y~w z are each = a a"bc(b'’ z c" z ), as 
at once appears from the identities 
ab.b' — c". a" c=a(b' 2 — c" 2 ), 
ab.c" — b'.ci/c = \b' 2 -c"f, 
a a .b —c .be— —b [b ~— c 
a'a".c —b'.bc= c' (IS 2 -c"~), 
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, + y 4, + z 4 + w 4 
„ I/-2 . 7,2 
® + C / oo 
6 /2 + c" 2 
/O . u 
«~ + cy „ „ 
ci"b 
/ O 9 i 9 9\ ^ I ^ / 9 9 I O 9 X I 1 / 9 9 I 9 9\ 
(x 2 y~+zhv~) — — f (xaz 2 -\-y~tir) + --(a^'+y-z-) 
2 (b'*- c "f K /-ab"c / n 
+ -- , L Vn ,, - xyzw= 0 ; 
a a bb cc 
or if we herein restore the c’s in place of the rectangular coefficients this is 
(• 4 _L n 4 
1 ~ °2 / 0 O, I 9 9\ 
vr 3 -(x-y-+cuw) 
r 4 i n 4 p 4 ! p 4 
4 o ^8 / ° o i o o\ | c 6 ■ L 9 / o 9 t o o\ 
■ (arer + y~w~) + (x~w~+y~z~') 
Oft c , 
C/Cfi 
6 u 9 
C 1 C 2 C 4 C 6 C 8 C 9 
which is the equation of the 16-nodal quartic surface. 
Substituting for x, y, z, iv their values d 4 , d 7 , d 8 , 9- ll (u), we have the equation con- 
MDOCCLXXX. 6 L 
