322] OK THE SKEW SURFACE OF THE THIRD ORDER, 
or, what is the same thing, 
nx — my — Bx + Ay = 0 : 
and when it meets the line (X — a = 0, Y — ft = 0), we have 
a = x + AZ, 
ft = V + BZ; 
and thence 
B {x — a) — A (y — /3) = 0. 
We have thus the system of equations 
(a 3 + /3 s ) xy — {a? + if ) a/3 = 0, 
X = x + AZ, 
Y=y + BZ, 
nx — my — Bx + Ay — 0, 
B (x - a) - A (y - ft) = 0 ; 
from which, eliminating (A, B, x, y), we obtain the equation of the surface. 
Writing in the last equation 
B = s (x — a), A = s (y — ¡3) 
(values which give Bx — Ay = — s (ftx — ay)\ we find 
X + asZ = (1 + sZ) x, 
Y + fisZ = (1 + sZ) y, 
(n + fts) x — (m + as) y = 0; 
whence also 
(:n + fts) (X + asZ) — (m + as) (Y + ftsZ) = 0, 
that is 
nX — mY + (na — m/3) sZ + s {/3X — ctY) = 0 ; 
or eliminating 6- from this equation and the two equations 
x — X + Z (x — a) s = 0, 
y - Y + Z(y- ft) s = 0, 
we have 
{{na — mft) Z + (3X — aY]{x — X) — Z (x — a) (nX — mY) = 0, 
{{na — mft) Z + ¡3X — aF} {y — F) — Z {y — /3) {nX — mY) = 0; 
these give 
nx = X {(na - mft) Z + ftX -aY}-aZ {nX - mY) 
= - mZftX + X {13X - a F) + mZa Y 
= {X-mZ){/3X-aY),