526] 
401 
526. 
ON A SURFACE OF THE EIGHTH ORDER. 
[From the Mathematische Annalen, vol. iv. (1871), pp. 558—560.] 
I reproduce in an altered form, so as to exhibit the application thereto of the 
theory of the six coordinates of a line, the analysis by which Dr Hierholzer obtained 
the equation of the surface of the eighth order, the locus of the vertex of a 
quadricone which touches six given lines. 
I call to mind that if (a, /3, y, 8), (a', /3', y', 8') are the coordinates of any two 
points on a line, then the quantities (a, b, c, f, g, h), which denote respectively 
W — ¡3'ry, iya — y'a, a/3' — a'/3, a8' — a'8, /38' — /3'8, y8' — y'8), 
and which are such that af+bg + ch = 0, are the six coordinates of the lineC). 
Consider the given point (x, y, z, w) and the given line (a, b, c, f g, h), and write 
for shortness 
P — hy — gz + aw, 
Q — — hx +fz + bw, 
9 X ~fy + cw > 
S = — ax — by — cz , 
then taking (X, Y, Z, IF) as current coordinates, the equation of the plane through 
the given point and line is 
PX+QY+RZ + SW=0. 
Considering in like manner the given point (x, y, z, w) and the three given lines 
(a lt b x , c u f lt g 1} /¿i), (a 2 >•••)> (a 3 ,...), then we have the three planes 
P x X + Q 1 Y+R 1 Z + S x W=0, 
P,X + Q 2 Y+R 2 Z+S 2 W=0, 
P 3 X + Q 3 Y+R 3 Z + S 3 W = 0, 
1 Cayley, “ On the six coordinates of a line,” Gamb. Phil. Trans, vol. xi. (1869), [435], pp. 290—823. 
c. viii. 51