63 
GENERAL EQUATION OF A CUBIC SURFACE. 
Hence, the number of such forms of equation 
= ^ X the number of closed hexagons 
= ■^.720 = 120 .^^ 
§ 18. In the case of a double two, 
the planes a, c, and b, d, are both triple tangent planes. 
h 
a 
* 
0 
d 
The intersection of these planes has clearly four points on the surface ; it is, there¬ 
fore one of the twenty-seven lines. 
Hence, for each line on the surface there are 5.4/1.2 = 10 pairs of triangles, each 
of which gives a double two. But if we reckon the two figures 
b 
• 
d 
• 
a 
* 
c 
d 
a 
* 
0 
h 
which represent the same set of four lines if they are double twos, as distinct double 
twos ; we say the number of double twos 
= 27 . 10 . 2 = 540. 
From a double three we can obtain three double twos; this is seen at once, for in 
a double three, such as 
C ; • * # 
^ I ^ 
I 
a j # 
d ti f 
we can leave out either of the pairs a, f; h, e; or c, d; and from Figure C, we see at 
once that we can from a double two form four double threes. 
Hence the number of double threes 
= 4/3 X the number of double twos 
= 4/3 . 540 = 4 . 180 = 720. 
* This iminber is given iu Salmon, ‘ Solid Geometry,’ 3rd edition, p. 466. 
