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704]
A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTIONS.
549
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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 -