258 PROFESSOR M. J. M. HILL ON THE LOCUS OF SINGULAR POINTS 
The coefficient of Y„ is 
Uu 
w 
0 
u 
WY, + YW, 
uw,+wu. 
UY,+YU, 
YW, 
UW + UW.-Ya 
YU, 
-i ks 
+ 
0 
— V 
0 
u 
w 
V 
u 
w 
Y 
U u 
WO u 
vw, u,w+uw-Ya yu, 
0 W 0 
- f- (VWV,+V J W,-«UV,-«VU,) 
U tt 
W (WVD, - UVW.) - ~ (V ! W., ; - uVU,) 
Uu 
= — {U, (uv - W 3 ) Y + W, (U W - vV) V} = 0. 
The coefficient of W y is 
JY 
Uu 
YU 0 
VW, U,W + UW,-Y,ti YU, 
u W Y 
WY + YW, UW, + Wffi UY + VU., 
+ 
Uu 
Y 
u 
U 
W 
0 
V 
= £ (U,W - V„„) + A (UWV, - YWUj.) 
Uw 
V 
= -{Y(UW-Y)) = o. 
Hence S 2 A/9a? 9y = 0. 
Hence all the differential coefficients of the second order vanish. 
Hence A contains E 3 as a factor. 
Example 15.— Envelope Locus, the parameters of both the Surfaces having Conic 
Nodes being infinite. 
Let the surfaces be 
z 3 a 3 + 2 + {ax b + fif = 0. 
(A.) The Discriminant, 
This is 
z 2 + ad x xg 
x 1 y 
*y y y~ + 2 
