450 
ON THE TRIPLE ^-FUNCTIONS. 
[702 
viz. for the points of the cubic hexad we have 
a \/bcf+ /3 Vcag + y Vabh + 8 V fgh = 0, 
ancl hence the condition in order that the four points (x lf y 1} z x ), (x 2 , y 2 , z 2 ), ( x 3> y-n 3,)» 
(x 4 , y 4 , z 4 ) may belong to the cubic hexad is 
VciGq#!, 
ctj)ihi y 
^ fig A 
v b 2 c 2 f 2 , 
Vc 2 a 2 g 2 , 
Va 2 b 2 h 2 , 
V fgA 
^b 3 C 3 f 3 , 
Vc 3 a 3 g 3 , 
V a 3 b 3 h 3 , 
ffgA 
V&4C4/4, 
\Zc 4 a 4 g 4 , 
VaJoJii, 
f fgA 
viz. we have thus the form of the determinant il which belongs to a cubic hexad. 
It is to be observed that the equation 
faf+ fbg + fch = 0 
remains unaltered by any of the interchanges a and f b and g, c and h; but we 
thus obtain only two cubic hexads; those answering to the equations 
a Vbcf + ¡3 Vcag + 7 Vabh + 8 V fgh = 0, 
and 
a fagh + /3 fbhf + 7 Vcfg + 8 Vabc = 0, 
which give distinct hexads. The whole number of ways in which the equation of the 
quartic can be expressed in a form such as 
Vo/‘+ \/bg + Vc/i = 0, 
attending only to the pairs of bitangents, and disregarding the interchanges of the 
two bitangents of a pair, is = 1260, and hence the number of forms for the determ 
inant il of a cubic hexad is the double of this, =2520, which is =36 x 70: but 
the number of distinct hexads is = 36, and thus there must be for each hexad, 
70 equivalent forms. 
To explain this, observe that every even characteristic except , and every odd 
characteristic, can be (and that in 6 ways) expressed as a sum of two different odd 
characteristics; we have thus (see Weber’s Table I.) a system of (35 + 28=) 63 
hexpairs; and selecting at pleasure any three pairs out of the same hexpair, we have 
a system of (63 x 20=) 1260 tripairs; giving the 1260 representations of the quartic 
in a form such as 
V af+ \/bg + VcA = 0. 
Each even characteristic ^not excluding qqq^ can be in 56 different ways (Weber, 
p. 23) expressed as a sum of three different odd characteristics, and these are such 
that no two of them belong to the same pair, in any tripair; or we may say that 
each even characteristic gives rise to 56 hemi-tripairs. But a hemi-tripair can be in 
5 different ways completed into a tripair; and we have thus, belonging to the same