262 
PROFESSOR M. J. M. HILL ON THE LOCUS OF SINGULAR POINTS 
The next six terms being obtainable from these last six by interchanging x and z 
vanish. 
The remaining six vanish in like manner. 
Hence all the third differential coefficients of A vanish. 
Hence A contains B 4 as a factor. 
Example 16. —Locus of Binodal Lines, 
Let the surfaces be 
z 2 {a 2 + <£ (x, y, z)} — (ax + b + yf = 0. 
(A.) The Discriminant. 
This is 
z 2 — x 2 — x — xy 
— x — 1 — y 
- - y - y~ + 2 V> ( x > z ) 
= - zty (x, y, z). 
(B.) Die Locus of Binodal Lines is Z — 0. 
For let £, 7), £ be any point on both the loci z = 0, ax -f- h + y = 0 . 
Then put x = £ X, y = y Y, z = £ + Z, so that £ = 0 , ai; + h + y = 0 . 
Therefore 
z? (a* + $ (l V. o + x + Y ^ + Z | + . . .) - (af + b + v + aX + Yf = 0. 
Hence the lowest terms in X, Y,Z are 
Z*{a* + *(£i 7 , £)} - («X + Y ) 2 = 0 . 
These break up into two factors. 
Hence the point ££ 17 , £ is a binode on the surface. 
Hence the straight line z = 0 , aa?-|-?> + ?/= 0 isa binodal line on the surface. 
And z = 0 is the locus of binodal lines. 
Hence the factor z 4 of the discriminant is accounted for. 
(C.) The Locus <f> (x, y, z) = 0 is connected with a Curve Locus, not a Surface 
Locus, of Ult mate Intersections . 
For the fundamental equations are in this case 
{a 2 + <f> (x, y, z)} — (ax + b + yf = 0, 
2z 2 a — 2* (ax -f b + y) = 0, 
2 (ax + b + y) = 0. 
z 
