44 MR. H. M. TAYLOR ON A SPECIAL FORM OF THE 
then 
T - K = L, T - L = K, 
T-M = N, T-N = M, 
it follows that the equations (C) satisfy equation (B) identically. 
Now the equations (C) are ecjuivalent to the equations 
I, - P = K- Q = 0' 
M-S = N-R = 0 I 
Hence the straig'ht line represented by these ecj[uations lies on the surface. 
Similarly we see that the pairs of equations 
T = K + m1 t = k + n' 
> and > 
T = L + NJ T = L + M 
also satisfy equation (B) identically. Hence the straight lines, whose equations are 
M-P=K-R=0 
N -Q = L -S =0 
and 
N-P = K-S =0l 
M~Q = L —R = 0j 
lie on the surface. 
We have thus the equations of fifteen straight lines which lie on the cubic surface 
represented by equation 
T3 _ T3 (K + L + M + N) + T (KL + km + KN + LM + MN + NL) 
- (KLM + LMN + MNK + NKL) = 0. (B). 
§ 2. Now, for convenience, let us take x, y, z, u instead of K, L, M, N, f.e., let us 
choose the tetrahedron ABCD formed by the four planes K, L, M, N as the tetra¬ 
hedron of reference. 
Then we may represent the four planes P, Q, R, S by 
= aT, y = bT, z = cT, u = cH, 
where T = ax + /3;^ + ys + Sw, and where a, h, c, d, a, y, 8 are constants. 
