446 
[702 
702. 
ON THE TRIPLE ^-FUNCTIONS. 
[From the Journal fur die reine und angewandte Mathematik (Crelle), t. lxxxvii. (1878), 
pp. 190—198.] 
A quartic curve has the deficiency 3, and depends therefore on the triple 
^-functions: and these, as functions of 3 arguments, should be connected with functions 
of 3 points on the curve; but it is easy to understand that it is possible, and may 
be convenient, to introduce a fourth point, and so regard them as functions of 4 
points on the curve: thus in the circle, the functions cos u, sin u may be regarded 
as functions of one point cos u = x, sin u = y, or as functions of two points, 
COS U = XXy + yy 1 , sin u = xy 1 — X{y. 
And accordingly in Weber’s memoir “Theorie der Abel’schen Functionen vom Geschlecht 
3,” (1876), see p. 156, the triple ^-functions are regarded as functions of 4 points 
on the curve: viz. it is in effect shown that (disregarding constant factors) each of 
the 64 functions is proportional to a determinant, the four lines of which are 
algebraical functions of the coordinates of the four points respectively: the form of 
this determinant being different according as the characteristic of the ^-function is 
odd or even, or say according as the ^-function is odd or even. But the geometrical 
signification of these formulse requires to be developed. 
A quartic curve may be touched in six points by a cubic curve: but (Hesse, 
1855*) there are two kinds of such tangent cubics, according as the six points of 
contact are on a conic, or are not on a conic; say we have a conic hexad of points 
on the quartic, and a cubic hexad of points on the quartic. In either case, three 
points of the hexad may be assumed at pleasure; we can then in 28 different 
ways determine the remaining three points of the conic hexad, and in 36 different 
* See the two memoirs “IJeber Determinanten und ihre Anwendung in der Geometrie” and “Ueber die 
Doppeltangenten der Curven vierter Ordnung,” Crelle, t. xlix. (1855).