GENERAL EQUATION OF A CUBIC SURFACE. 
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§ 1. If K, L, M, N, P, Q, P, S be eight linear functions of point coordinates 
in three dimensions, so that any one of them equated to zero represents a plane, 
then the equation 
KLMN = ^PQPvS.(A) 
represents a c|uartic surface, which passes through each of the 16 straight lines given 
by the intersection of one plane from each of the groups, K, L, M, N and P, Q, R, S. 
The equation contains 3 X 8 + 1, or 25 available constants. 
Now if the planes be so related that the intersections of the pairs of planes K, P ; 
L, Q ; M, P; N, S, lie on a plane T, or, in other words, if the two tetrahedrons 
represented by the two sets of planes K, L, M, N and P, Q, P, S be in perspective, 
then, without further affecting the generality of the choice of the eight planes, we 
may assume 
K + P = L + Q = i\I + P = N + S = T; 
and the equation of the surface may be written 
KLMN = 0(T - K) (T - L) (T - M) (T - N). 
This is the equation of a quartic surface, which jjasses through 16 straight lines, 
and in which there are 3 X 5 + 4 + 1, or 20 available constants. 
If, fui’ther, we take 0=1, the term KLMN cancels, and the equation becomes 
divisible by T, the remaining factor equated to zero giving 
T-s _ (K + L + M + N) 
4- T (KL + KM + KN + LM + MN + NL) 
- (KLM + LMN + MNK + NKL) = 0.(B) 
the equation of a cubic surface, which passes through twelve straight lines. 
K, 
K, P(8) 
K, SOO) 
L, PW 
L, P(«) 
L, SOO 
M, P^^) 
M, QW 
• 
M, S02) 
N, 
N, Q(5) 
N, P(7) 
and which contains 19 
of a cubic surface. 
And since, if 
available constants, the full number for the general equation 
T = K + L 
T = M + N 
(i 
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