94 
ON THE SKEW SURFACE OF THE THIRD ORDER. 
[322 
or, observing that X = P + mZ, Y = Q + nZ, and thence 
PY- QX = Z (Pn - Qm), XP 2 + QY 2 = P 3 + Q 2 + Z(mP ! + nQ 2 ), 
the equation becomes 
(a 3 + /3 3 ) PQ + (a 2 P - pQ) (PY - QX) - s/3 (P 2 X + Q 2 Y) = 0, 
or, what is the same thing, 
(aP 2 - /3Q 2 ) (aY— j8X) + PQ (a 3 + /3 :; - a 2 X -/3 2 Y) = 0 ; 
whence, making a slight change in the form, and restoring for P, Q their values, the 
equation is 
{«(X - mZf - j3(Y- nZf) {a(Y — /3)- /3 (X - a)} 
- (X - mZ) (F - nZ) {a 2 (X - a) + /3 2 (Y- ¡3)} = 0, 
a form which puts in evidence as well the simple line (X — a. = 0, F — ¡3 = 0) as the 
nodal line (X — mZ = 0, Y—nZ — 0). 
If Z = 0, we have 
(«X 2 — /3F 2 )(aY — &X) — XY{a 2 (X - a) + /3 2 (F- /3)} = 0, 
which is in fact the cubic curve (a 3 + ¡3 3 )XY—a/3 (X s + F 3 ) = 0. 
Reverting to a former system of equations 
nx — my — Bx + Ay — 0, 
B(x — a) — A (y — /3) = 0, 
or, as these may be written, 
we find 
Bx — Ay — nx — my, 
Ba— Aß = nx — my, 
B (ßx - ay) = (ß - y) (nx - my), 
A (ßx — ay) = (a — x) (nx — my); 
so that we have 
x = x + (2-_^(ux-my) 
(3x — ay 
\r , (¡3 - y) (nx - my) ry 
r ~ y+ — ' 
as the equations of the generating line which passes through the point (x, y) of the 
cubic curve. 
2, Stone Buildings, W.C., October 28, 1862.