178 PROFESSOR M. J. M. HILL ON THE LOCUS OF SINGULAR POINTS 
It is therefore 
x z — cx + 2w + 2z — 1-= 0. 
e 
This does not contain the locus of conic nodes. 
It is an envelope touching f at the two points 
V = h a + ~- 
z = fi) -f- 1. 
Art. 4 .—Investigation of the conditions which are satisfied at any point on the Locus 
of Conic. Nodes. 
In the preceding article it was shown that the surfaces (3) have in general a curve 
locus of conic nodes. 
If, how'-ever, every one of the surfaces (3) has a conic node, then equations (3), (6), 
(7), (8) are equivalent to three independent equations only, and the locus of conic 
nodes is a surface, whose equation is obtained by eliminating a and h between any 
three of the four equations (3), (6), (7), (8). 
It will be proved that such a surface locus of conic nodes is a part at least of the 
locus of ultimate intersections. 
With the notation of the last article, equations (11)-(18) hold ; but now there is no 
relation between a and f3. 
There is a conic node on the surface 
f{x, y, z, a + 3a, /3) = 0.(24). 
Hence (15) must hold when S/3 = 0. 
Hence 
D//Da = 0.(25). 
Similarly 
D//D£=0.(26). 
Since (11), (25), and (26) hold at all points of the conic node locus, it follows that 
the conic node locus is a part, at least, of the locus of ultimate intersections. 
The position of the tangent plane to the conic node locus may be obtained from 
(16)-(18) by eliminating 8a, S/3 ; and then using the relations 
