150 
PROFESSOR M. J. M. HILL ON THE LOCUS OF SINGULAR POINTS 
Take « = <£(£ 77, Q and consider the single surface 
[<t> fa y, z) - <f> (£ y, i)J + x ( x > y> z ) [V» (*» y> Z )J = °- 
Put *=f-fX,i/=))-fY 1 z = ^+Z. 
Then the lowest terms in X, Y, Z are 
X | + Y | + Z lj 
+ x (£ v> 0 
x }+4t+ zS C 
This breaks up into two factors of the first degree in X, Y, Z. 
Hence £ 77, £ is a binode on the surface considered. Now the only relation 
satisfied by £ 77, £ is «/»(£ 77, £) = 0. 
Hence any point £ 77, £, on the surface if, (x, y, z ) = 0, is a binode on 
[cf>(x, y,z) - <f> (£ 77, £)] 2 + X y, z ) y, z)J = 0 . 
Hence every point of intersection of the surfaces 
«/» y> z ) = o, 
and 
[<£ (®, y> z ) - a J + x (*» y> z ) ( x > y> z )l = °> 
is a binode on the latter surface. 
The equations of the binodal line of this surface are, therefore, 
if, (as, y, z) = 0 , 
<£ (x, y, z) = a. 
This accounts for the occurrence of the factor [1 fj ( x, y, z)f in A. 
C. The Envelope Locus is y (x, y, z) = 0. 
This may be proved as in Example 1. 
Art. 6 .—Examination 0 / the Discriminant A, and its Differential Coefficients, when 
a Surface Locus of Unodal Lines exists. Proof that A contains U 3 as a factor. 
Differentiating ( 7 ) with regard to x and y, 
__ f S^Q 0Q fDf Lf Baf 
Ox 1 ^ 1 Bx 2 Bx \ Dm Da, Bx) " 3_ 
Did > 
Da’ Da, Bx j 
(41), 
