526] 
ON A SURFACE OF THE EIGHTH ORDER. 
403 
where the symbol stands for the determinant 
But as is well known this equation may be written 
(*) 
(126)(346)(145)(235) - (146)(236)(125)(345) = 0, 
where (126) etc. denote the determinants 
Pi> Qi> Pi etc., 
P„, Q 2 , r 2 
p„ Qz> R 3 
or, what is the same thing, they denote the functions above represented by the like 
symbols (126) = (ayg 2 h 6 ) of + etc. The equation (*) just obtained is Hierholzer’s equation 
for the surface of the eighth order, the locus of the vertex of a quadricone which 
touches six given lines. 
I remark that in my “ Memoir on Quartic Surfaces,” Proc. Lond. Math. Soc. vol. ill. 
(1870), [445], pp. 19—69, I obtained the equation of the surface under the foregoing 
form [123456] = 0 or say [(P, Q, P) 2 ] = 0, noticing that there was a factor w 4 , so that 
the order of the surface is = 8; and further that the equation might be written 
w 8 exp. ^ [x (gd c - hd b ) + y (hd a - fd c ) + 0 (fd h - gd a )} [(a, b, c) 2 ] = 0, 
where exp. © (read exponential) denotes e®, and [(a, b, c) 2 ] denotes 
by", Cy v byCy, Cy ay, ayby 
a 2 2 ,.. 
Also that the equation contains the four terms 
¿r 8 [(a, - h, gf] + y 8 [(h, b, -/) 2 ] + ¿r 8 [(- g, f c) 2 ] + w 8 [(a, b, c) 2 ] = 0. 
Cambridge, 12 September, 1871. 
51—2