168 
PROFESSOR M. J. M. HILL OR THE LOCUS OF SINGULAR POINTS 
Let f = U« 2 + 2Ya + W = 0, where U, V, W are rational integral functions 
of x, y, z. 
Then the conditions f— 0, D/’/Da = 0, D 2 //D« 2 — 0 are equivalent to 
Ua 2 + 2Va +W = 0, 
U a + V =0, 
U =0, 
at all points of the locus of ultimate intersections. 
Hence U = 0, V = 0, W = 0 at all such points. 
Hence unless IT, V, W have a common factor, which could in that case be removed 
from the equation f — 0, the locus of ultimate intersections is not a surface, and hence 
its equation cannot be obtained bj equating a factor of the discriminant to zero. 
Hence this case need not be further considered. 
Art. 11 .—If the surface f(x, y, z) = 0 have upon it a curee at every point of which 
there is a conic node, then the tangent cones at the co?iic nodes must break up 
into two planes* 
Let £, 7 ), £; f + ££ y 4* Sy, £ + S£ ; be neighbouring points on the curve. 
Then since there is a conic node at y, Cl 
M V. £) = 0, D//Df = 0, D//D, = 0, D//D£ = 0. 
And since there is a conic node at £ -f- S£, y -f Sy, l + §£, four other equations hold, 
which by means of the above give 
k a w + h, y] (w 2 + k a m* 
+ 2h, a (&?) (SO + 2[c, fl (80 (§i) + 2[i, y] (so (S v ) + . . =0, 
Kf](8f) + [^]W + KCl(8£)+ ..... =0, 
[yi £] (§£) + [y> y] (°y) + [y, £] (H) +.= °> 
fl (§0 + [£ (&?) + [£, a (§0 +.= 0 . 
Retaining only the principal terms in the last three equations, it follows that 
* The geometry of a surface of continuous curvature shows at once that there cannot be a curve of 
conical points on a surface. 
