AND LTNES IN THE INTERSECTIONS OF A SYSTEM OF SURFACES. 177 
(A.) The locus of conic nodes is the straight line. 
x = \c, V + 2 = £ c 3 - 
To find this locus it is necessary to eliminate a, h between (19) and 
^ — 2x — c = 0 .(20), 
Vx 
‘2e(y- | a) = 0 .( 21 ), 
g= 2 ( 2 -i 6 ) = 0 .( 22 ). 
Therefore 
x = \ c, y — \a, z — \b, y + z — \c 2 . 
Hence the locus of ccnic nodes is the straight line, 
x = ^c, y + z = ic 2 .(23). 
(B.) The locus of conic nodes lies on the general integral of the partial differential 
equation of the surfaces (19) obtained by putting b — \ c 2 — a. 
To determine this general integral take the values of x, y, z from ( 20 )-( 22 ), and 
substitute in (19). This gives a + b = ^ c 3 . 
Hence the general integral is obtained by eliminating a from 
a ; 2 + e (y — i «) 3 + (z + 4 a — | c 2 ) 2 — cx + \c 2 = 0 , 
and 
“ e (y - \ a) + (z + £ a — | c 2 ) = 0 . 
Hence it is 
e {y + 2 “ i C ~Y + (1 + e) (x - \ c) 2 = 0 . 
It contains the locus of conic nodes, whose equations are given in (23). 
(C.) The locus of conic nodes does not lie on the locus of ultimate intersections. 
For the equation of the locus of ultimate intersections is obtained by eliminating 
a, b between 
f — x 2 e {y — \ a) 2 + (2 — ^ b) 2 — cx + a + b = 0 , 
Df 
Da = - e (y - £ ®) + 1 = 0, 
g=-(*-H)+i = o. 
2 A 
MDCCCXCII.—A. 
