GENERAL EQUATION OF A CUBIC SURFACE. 69 
A set of lilies such that they form triangles when read in rows or columns, as 
a, h, 0 
f 
is obtainable from everyone of the possible forms of the equation of the cubic surface, 
such as LMN = PQR. 
There are 120 such sets (§ 17), and when one is chosen there is only one way of 
completing the set of triangles by similar sets of nine lines (§ 20). Therefore, there 
must be 40 different ways in which all the lines on the surface can be arranged, 
sucli as 
a h c 
j k 1 
s t u 
d e f 
m n 0 
V IV X 
9 
p q r 
g z io 
such that each row and each column of any one of the three sets gives a triangle. 
The number may also be calculated by considering how many such sets as 
a b c , 
d e f 
9 ^ 
exist containing a definite line a. 
There are five triangles which contain a. There are, therefore, ten pairs of such 
triangles, or ten selections of a, h, c, d, g in the set ; for each pair of h and d tliere are 
four lines which could take the place of c, and then the set is determined uniquely. 
Tliere are, therefore, 10X4 = 40* such sets. 
* Stukm, ‘ Synth. Enters, iiber Fliicheu Dritter Ordnaug. 
