152 PROFESSOR M. J. M. HILL ON THE LOCUS OF SINGULAR POINTS 
Example 3.— Locus of Unodal Lines. 
Let the surfaces be 
[<f> ( x > y > z ) - a T + [x ( x > y> Z )J = o. 
(A.) The Discriminant. 
The discriminant is found by eliminating a between the above and 
— 2 [<f> (x, y , z) — a] = 0. 
Hence it is 
[x ( x > y> z )J- 
Hence the locus of ultimate intersections is 
lx ( x > y> Z )J = °- 
(B.) The Locus of Unodal Lines is y (x, y, z) = 0 . 
Let f y, £ be any point on the locus y (x, y, z ) = 0 . 
Let a — <f> (£, y, £), and consider the single surface 
[(f) (x, y, z) — (f) (£, y, £)] 3 + [y (x, y, z)J = 0. 
Put a? = £ + X, y = y - p Y, z = £ -}- Z ; then the lowest terms in X, Y, Z ar 
d(j) 
dr) 
+ Z 
d(j) 2 
' 
Hence y, £ is a unode on the surface 
[(f) (x, y, z) — (f) (ij, y, £)] 3 + [x y> Z )J = 0 . 
Lienee y (x, y, z) = 0 is the locus of unodal lines on the surfaces 
[(f) (x, y, z) — aj 4- [y (x, y, z)] 3 = 0 . 
The unodal line on any one of the surfaces is given by the equations 
y (x, y, z) = 0 , 
(f) (x, y, z) = a. 
This accounts for the occurrence of the factor [y ( x, y, z)] 3 in the discriminant. 
