176 
PROFESSOR M. J. M. HILL ON THE LOCHS OF SINGULAR POINTS 
and f -f- S£, y + &y, £ -f S£ the coordinates of the conic node on the surface 
f(x, y, z, a + 8a, (3 + 8/3) = 0.(10), 
then the following 1 equations all hold at the same time, 
/(£ V, l a, (3) = 0.(11), 
D /(£ v> & a > ft) _ a /io\ 
p/(g , V, K, «, ft) 
T 
(13). 
D/(f, 1, f. «, ft 
D? 
• (14), 
and four other equations, which, by means of the above, become 
§£ (s«> + H m = o.(is), 
[?, fl(Sf) + [f, ,](&,) + [f, £](§{) +[f, a](Sa) + [f, /3](8/3) = 0 . (16), 
h> f](8f) + fo> >)](8i7) + fo, (1(S£) + [>), a] (8a) + [j), /3](S£) = 0 . (17), 
[£, fl (Sf) + [£, ,] (8,) + [£, (] (8Q + [{, a] (8a) + [£, /3] (8/3) = 0 . (18). 
If 8 a, 8/3 be eliminated from (15)-(18), the ratios : 817 : 8 £ are determined. These 
ratios determine the tangent line to the curve locus of conic nodes. 
If /3 be determined as a function of a, so that (11)—(14) can be satisfied by the 
same values of y, £, then the equations (11) and (15) show that the locus of conic 
nodes is a curve lying on one of the general integrals of the partial differential 
equation of the surfaces (3). 
Example 1 . Curve Locus of Conic Nodes. 
Let the surfaces be 
x 2 + e (y — ^ of + (2 — ^ 8) 2 — cx + a + b = 0 . . . . (19), 
where e, c are fixed constants; a, b the arbitrary parameters. 
