503] 
THE VERTICES OF CONES WHICH SATISFY SIX CONDITIONS. 
135 
and if the equations of the lines 8, e are as before {x = 0, y = 0) and {z = 0, w = 0); 
then for the line in the tangent-plane meeting the lines 8, e, we have 
%x + rjy = 0, £z + cow = 0. 
These last equations may be represented by 
f =ly, 7] = —lx, £= mw, co — — mz; 
and, substituting these values, we have 
{A,...'^ly, —lx, mw, — mzf = 0, 
{a , ..-5}y, ~ mw > ~ mz Y = 0, 
that is 
{Ay 1 — 2Hxy + Bx 1 , — Fxw + Gyw — Lyz + Mxz, Gw 2 — 2Nwz + Dz 2 $7, to) 2 = 0, 
and 
{ay — fix, yw — 8z\l, m) = 0. 
Whence, eliminating l, to, we have the quartic equation 
{Ay 1 — 2Hxy + Bx 2 , — Fxw + Gyw — Lyz + Mxz, Gw 2 — 2Nzw + Dz^yw — 8z, fix — ay) 2 = 0. 
Further Investigation as to the Surface abafiy8. 
59. The theorem that in the surface abafiy8, the equation of which is 
Norm {fpaa.pba.p 2 fiy8 - fpafi .pbfi. p 2 y8a +Vpay .ply .p 2 8afi - fpa8 .pb8 .p 2 afiy] = 0 ; 
the lines {ab, a, fi, 7) are tacnodal, each sheet touching along the line the quadric 
p 2 afiy, may be proved in a different manner by investigating the intersection of the 
surface with the quadric p 2 afiy. 
For this purpose take the equation of the quadric to be yz — xw = 0 ; the equations 
of the lines a, fi, y will be 
/z — \ a w = 0\ /z-\pw=0\ iz-\ y w= 0\ 
\x - \ a y = 0/ \x-\fsy = 0/ ’ \x - \ y y - 0/ ’ 
and we may write {a, b, c, f, g, h) for the coordinates of the line 8. The equation of 
the surface will be 
Norm -|X [± ^paa .pba (\p — \ y ) 
{a —f ) xz — + \y) yz + \p\ y yw 
+ (b — g) (y z ~ xw ) 
+ c{z — \pw) {z — \ y w) 
+ h (x— \py ) {x — A y z ) 
— \/pa8 . pb8 (A,3 — Ay) (Ay — A tt ) (A a — \p) {yz — xw)| ;