[704 
704] 
A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTIONS. 
549 
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locus 
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space, 
exad : 
surface, which is, in fact, a Rummer’s 16-nodal quartic surface. For if for a moment 
x. y and z. w are two pairs out of a tetrad, and r.s be either of the remaining 
pairs of the tetrad; then we have rs a linear function of xy and zw: squaring, r 2 s 2 
is a linear function of x 2 y 2 , xyzw, z 2 w 2 ; but we then have r 2 and s 2 , each of them 
a linear function of x 2 , y 2 , z 2 , w 2 ; or substituting we have an equation of the fourth 
order, containing terms of the second order in {x 2 , y 2 , z 2 , w 2 ), and also a term in 
xyzw. It is clear that, if instead of r.s we had taken the remaining pair of the 
tetrad, we should have obtained the same quartic equation in (x, y, z, w). And 
moreover it appears by inspection that, if xy and zw are pairs in a tetrad, then xz 
and yw are pairs in a second tetrad, and xw and yz are pairs in a third tetrad: 
we obtain in each case the same quartic equation. We have from each tetrad of 
pairs six sets of four functions (x, y, z, w): and the number of such sets is thus 
(T6.30 =) 60: these are shown in the foregoing “ Table of the 60 Gopel tetrads,” viz. 
taking as coordinates of a point the four functions in any tetrad of this table, the 
locus is a 16-nodal quartic surface. 
122. To exhibit the process I take a tetrad 4, 7, 8, 11 containing two odd 
functions; and representing these for convenience by x, y, z, w, viz. writing 
A 4 , A 7 , A 8 , % 1 (u) = x, y, z, w, 
we have then X, Y, Z, W linear functions of the four squares, viz. it is easy to 
obtain 
a (x 2 + z 2 ) — 8 (y 2 + iv 2 ) = 2 (a 2 — S 2 ) X, 
S( „ )-a( „ ) = 2( „ )W, 
— /3 (x 2 — z 2 ) + 7 (y 2 — w 2 ) = 2 (/3 2 — 7 2 ) Y, 
-7< » ) + £( „ ) = 2( „ )Z. 
Also considering two other functions A 0 (w) and or as f° r shortness I write 
them, A 0 and % 2 , we have 
A 0 2 = aX + ¡3 Y + 7 Z + 8 W, 
A 12 2 = aX — ft Y — yZ + 8W, 
and substituting the foregoing values of X, Y, Z, W, we find 
MX 2 = Ax 2 + By 2 + Cz 2 + Dw 2 , 
M% 2 = Cx 2 + By 2 + Az 2 + Bio 2 , 
four 
where, writing down the 
values first in terms of a, 
/3, 7, 8 and 
3S, or 
the c’s, we have 
M = 
(a 2 - 8 2 ) ((3 2 - 7 2 ) = i 
• c s 4 c 4 4 , 
A = 
/3 2 8 2 - oi 2 y 2 = „ 
- C 2 2 c 6 2 , 
B = - 
aB (/3 2 - 7 2 ) + fiy (a 2 - B 2 ) = „ 
P 2p» 2 ___ p 2p» 2 
G*3 o 4 0 15 t/ 8 , 
-etrad 
G = 
II 
V 
1 
a 
Ci 2 C 9 2 , 
lartic 
D = - 
aB (/3 2 — 7 2 ) — /3y ( a ' 2 — = » 
Ci 5 “C 4 2 C 3 -C 8 -