448
ON THE TRIPLE ^-FUNCTIONS.
[702
Hence the condition in order that the four points (x lf y lt zj, (x 2 , y 2 , z 2 ), (x 3 , y 3 , z 3 ),
(x i} Vi, 2 4 ), assumed to be points of the quartic, may belong to the conic hexad, may
be written
^ fAgi,
aq Vcq,
Vi
z x ^a 2
u
O
©
II
^ fi°A >
x 1 Vaq,
2/i Vaj,
z l \/a 1
VfAg2,
y^a 2i
z 2 V a 2
fc 2 h 2 ,
x 2 a 2 ,
z 2 V a 2
X 3 V& 3 ,
2/3 Va 3 ,
z 3 \/a 3
\/ f 3 c 3 h 3 ,
x 3 Va 3 ,
2/3 Va 3 ,
z 3 V a 3
V/4%4,
aqVa 4 ,
2/4 V« 4 ,
ZiVCLi
V/ 4 cA,
x 4 Va 4 ,
2/4 Va 4 ,
z x va 4
where, as before, the x, y, z may be replaced by any three of the letters a, b, c,
f g, h, or by any other linear functions of (x, y, z): and, moreover, although in
obtaining the condition we have used for the quartic the equation
Va/+ Vbg + Vch = 0,
depending upon six bitangents, yet from the process itself it is clear that the condition
can only depend upon the particular bitangent a = 0: calling the condition il = 0, all
the forms of condition which belong to the same bitangent a = 0, will be essentially
identical, that is, the several determinants H will differ only by constant factors; or
disregarding these constant factors, we have for the bitangent a = 0, a single determinant
fi, which may be taken to be any one of the determinants in question. And we
have thus 28 determinants il, corresponding to the 28 bitangents respectively.
Coming now to the cubic hexads, Hesse showed that the equation of a quartic
curve could be (and that in 36 different ways) expressed in the form, symmetrical
determinant = 0, or say
a,
h,
K
b,
9>
l =0,
f m
9, f c, n
l, m, n, d
where (a, b, c, d, f, g, h, l, m, n) are linear functions of the coordinates; and from
each of these forms he obtains the equation of a cubic
a, h,
h, b,
9> f
l, m,
9> l >
f, m >
a
= 0,
¡3
c, n, 7
n, d, 8
a, /9, 7, 8
containing the four constants a, /3, 7, 8, or say the 3 ratios of these constants,
touching the quartic in a cubic hexad of points: that the cubic does touch the
quartic in six points appears, in fact, from Hesse’s identity
