410] A THIRD MEMOIR ON SKEW SURFACES, OTHERWISE SCROLLS. 319 
let p be the third root, then we have 
6 + {f> + p ~ X' 
6$ + p (6 + (f>) = ^, 
p. n w 
^■P = Y , 
and thence 
8 + 'i> = j,(Y-pX), ^ = I(2-pF + p'X) = A W, 
(X, -r, z, - trip, iy=o. 
Substituting for 0 + <f> and 0<p their values in terras of p, we find 
Fp{Y 2 - ZX - pXY} - GpX(Y-pX) + HpX 2 
+ W {(.AX+BY+ GZ) -p(BX + GY) + p 2 GX) = 0, 
or, what is the same thing, 
p 2 X (GX - FY + GW) 
- p {F(Y 2 - ZX) — GXY + HX 2 - BXW-GYW} 
+ W(AX + BY+CZ) = 0; 
from which and the equation 
(X, — Y, Z, — vr$p, 1) 3 = 0, 
we have to eliminate p. 
70. Writing for shortness 
( ., H, - G, A\X, Y, Z, W) = a, 
(-H, F, Bl „ ) = /3, 
( G, -F, ., 0$ „ ) — 7> 
{-A, -B, -G, .$ „ ) = 8, 
and therefore aX + j3Y + yZ + 8W = 0 : the two equations are 
p 2 X7 + p (— yY — ¡3X) — 8 IF = 0, 
p*X-p 2 Y+pZ - W=0. 
Writing the first equation in the form 
ry(p 2 X-pY+Z)-/3(pX-Y) + <xX = 0, 
multiplying by — p, and reducing by the other equation, 
/3 (p 2 X — pY) — pa. X — <yW= 0, 
or, as this may be written, 
/3 (p 2 X — pY+Z) -a(pX — Y) — aY—/3Z — yW = 0.