PROFESSOR CAYLEY ON SKEW SURFACES, OTHERWISE SCROLLS. 
117 
Hence for the line which is the intersection of the two osculating planes 
(*, -1XX,Y, Z, W)=0, 
(f, -<p\ P, -1JX, Y, Z, W)=0, 
forming the expressions of the six coordinates, but omitting the common factor <p— 0, 
these are 
a, b, c,f, g, h=Q 2 +d<p+<p 2 , 1, dip, dip(d+p), ; 
we have thus between the parameters Q, <p the relation 
(F, G, H, A, B, CX& 2 +fy+f, -8-<P, 1, &<P(G+<P), 5 2 <p 2 )=0; 
and the equation of the scroll is obtained by eliminating 0, <p between this equation and 
the last-mentioned two equations satisfied by S, <p respectively. 
69. We see that 6, cp are two of the roots of the equation 
(X, -Y, Z, -WJw, 1) 3 =0; 
let g be the third root, then we have 
=x’ 
and thence 
«+?>=s(Y-gX), 
(X, -Y, Z, -Wft, 1)’=0. 
Substituting for S-j-i p and Q<p their values in terms of g, we find 
F ? {Y 2 -ZX-^XY}-G^X(Y- ? X)+H f X 2 
+W{(AX+BY+CZ)- § (BX+CY)+ f 2 CX}=0, 
or, what is the same thing, 
g> 2 X(GX— • FY -f CW) 
— g> { F( Y 2 — ZX) — GX Y -f HX 2 — BX W — C Y W } 
+W(AX+BY+CZ)=0; 
from which and the equation 
(X, -Y, Z, -Wig, 1)*=0, 
we have to eliminate g. 
