162 PROFESSOR M. J. M. HILL OR THE LOCHS OF SINGULAR POINTS 
Example 5.— Locus of Binodal Lines which is also an Envelope. 
Let the surfaces be 
[</> {x, y, z)J + (f) (x , y, z) [i jj ( x , y, z ) — a] + [xjj ( x, y, z) — a ] 3 = 0 . 
(A.) The Discriminant. 
The discriminant is found by eliminating a between the above and 
<f> {sc, y, z) + 3 [xfj (x, y, z ) — cif = 0. 
The last equation gives 
*// {sc, y, z) — a = ± y{ — i <f> {sc, y, z)). 
Hence the eliminant is 
{[(f) {x, y, z)J + | f {x, y, z) y - ± f (x, y, z)} 
X {[(f) {x, y, z)J - f <j) (x, y, z) x/ - ± $ (x, y, z)) 
= W> {d y, 2 )] 3 [f {sc, y, z ) + -if]. 
Hence the locus of ultimate intersections is 
[(f) {x, y, z)J [f {x, y, z) + 2 V] = 0 . 
(B.) The locus of Binodal Lines {which is also an Envelope) is f{x, y, z) = 0 . 
For let 7], l be any point on cf) {x, y, z) = 0. 
Take a = \f> {£, y, £), and consider the single surface 
[f {x, y, z)J + <f> {sc, y, z) [xfj {x, y,z)-xp(g, y, £)] + [xfj {x, y, z) - xfj {£, y, f )] 3 = 0 . 
Put a? = £ -j- X, y = y Y, z = £ -\- Z ; then the lowest terms in X, Y, Z are 
X | + Y 8-+ Z | 
+ Yd + zf + (x^ + Y ? A + zt£)l. 
07] c£J V of 07 o?/ J 
Hence the origin is a binode. 
Hence (f> {x, y, z) = 0 is the locus of binodal lines. 
Further, because the biplanes are, when the origin is at the binode, 
x gf + Y fi + Z ff = 0 ’ 
X | + Y | + Z |) + ( X | + Y | + Z | 
= 0 . 
