702] 
OX THE TRIPLE ^-FUNCTIONS. 
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ways the remaining three points of the cubic hexad: or what is the same thing, 
there are 28 systems of cubics touching in a conic hexad, and 36 systems of cubics 
touching in a cubic hexad. The condition in order that four points of the quartic 
curve may belong to a hexad (conic or cubic) is given by an equation 0 = 0, where 
O is a determinant the four lines of which are algebraical functions of the coordinates 
of the four points respectively: but the form of such determinant is different according 
as the condition belongs to a conic hexad, or to a cubic hexad: we have thus 28 
conic determinants and 36 cubic determinants, O; and the 64 ^-functions are pro 
portional to constant multiples of these determinants; viz. the odd functions correspond 
to the conic determinants, and the even functions to the cubic determinants. 
First, as to the conic hexads: the points of a conic hexad lie in a conic with 
the two points of contact of some one of the bitangents of the quartic curve: so 
that, given any three points of the hexad, these together with the two points of 
contact of the bitangent determine a conic which meets the quartic in the remaining 
three points of the hexad. Suppose that a, 6, c, f g, h are linear functions of the 
coordinates such that the equation of the quartic curve is 
V«/ + \/bg 4- (ch = 0 ; 
then a = 0, 6 = 0, c = 0, /=0, g = 0, h= 0 are six of the bitangents of the curve, 
and the bitangent a = 0 touches the curve at the two points of intersection of this 
line with the conic bg — ch = 0. The general equation of a conic through these two 
points a — 0, bg — ch = 0, may be written 
bg — ch +a (Ax 4- By + Gz) = 0, 
where for x, y, z we may if we please substitute any three of the six linear functions 
a, 6, c, f g, h, or any other linear functions of the coordinates (x, y, z): and the 
equation may also be written 
af ± (bg — ch) 4- a (Ax 4- By + Cz) = 0. 
Adopting this latter form, and considering the intersections of the conic with the 
quartic, that is, considering the relation 
(af+(bg + (ch = 0 
as holding good, we have 
af+ bg — ch = — 2 (afbg, 
cif—bg + ch = — 2( afch, 
and we thus have at pleasure one or other of the two equations 
— 2 (afbg 4- a (Ax 4- By 4- Gz) = 0, 
— 2 Vafch 4- a (Ax + By 4- Gz) — 0, 
— 2 (fbg + (a (Ax 4- By 4- Gz) = 0, 
— 2 (fch 4- (a (Ax + By + Gz) = 0. 
that is,