92 
ON THE SKEW SURFACE OF THE THIRD ORDER. 
[322 
and 
fly = F {(wa — wifi) Z + ftX — aY] — ftZ (nX —mY) 
= nZaY + Y(ftX - aY) - nZftX 
= (Y — nZ) (ftX-aY), 
where 
il = (wa — m/3) Z + (ftX — aY) — Z (nX — mY) 
= ft (X — mZ) — a (Y — {w (Z — mZ) — m(Y — nZ)} 
= (/3 — wF) (Z — mF) — (a — mZ) (F—nZ); 
so that 
(Z — toF) (f3X — aY) 
X ~ nZ) (Z — mZ) — (a — mZ){Y — nZ) * 
(Y-nZ)(/3X-aY) 
y ~ (0 - nZ) (Z - mZ) - (a - mZ) (F - nZ) 5 
which equations give the coordinates {x, y) of the point in which the generating line 
through the point (Z, Y, Z) of the surface meets the cubic 
(a 3 + /3 3 ) xy — (& 3 + y 3 ) aft = 0. 
Substituting these values of (x, y) in the equation of the cubic, we obtain the equation 
(a 3 + ft 3 ) (X — mZ) (F — nZ) {(ft — nZ) (x — mZ) — (a — mZ) (F — nZ)\ 
— aft (ftX — aY) {(Z — mZ) 3 + (F — nZ) 3 } = 0 ; 
or, as it may be written, 
(a 3 + ft 3 ) (X — mZ) (F— nZ) {ft (X — mZ) — a (Y — nZ)\ 
+ (a 3 + ft 3 ) (X — mZ) (F— nZ) Z (mY—nZ) 
— aft (ftX — aY) {(Z — mZ) 3 + (F — nZ) 3 ] = 0. 
This equation contains, however, the extraneous factor 
ft (X — mZ) — a (Y — nZ), 
which, equated to zero, gives the equation of the plane through the node and the line 
(x — mz = 0, y — nz = 0). In fact, assuming 
(a 3 + ft 3 ) (X — mZ)(Y — nZ) Z (mY — nZ) — aft (ftX — aY) {(Z — mZf + (Y — nZ) 3 ) 
= {ft (X - mZ) - a(Y-nZ)} <P (X, Y, Z), 
it will presently be shown that <E> is an integral function. Hence, omitting the factor 
in question, we have 
(a 3 + ft 3 ) (X- mZ) (Y - nZ) + <t> (X, F, Z) = 0,