218 
PROFESSOR M. J. M. HILL ON THE LOCUS OF SINGULAR POINTS 
Now if each of the differential coefficients of A be formed up to the third order, then 
every term in the result will contain as a factor one of the following quantities :— 
fi or f -2 or a first differential coefficient of f or f 2 . 
Hence, wdien x = £ y = 77, 2 = £, A and all its differential coefficients up to the 
third order vanish. 
Hence, if B = 0 be the equation of the binode locus, such that the edge of the 
binode always touches the binode locus, A contains B 4 as a factor by Art. 1, Pre¬ 
liminary Theorem B. 
Example 10. —Locus of Biplanar Nodes such that the Edge of the Biplanar Node 
always touches the Biplanar Node Locus. 
Let the surfaces be 
a (x — a ) 3 + f3 (y — 6) 3 + 3 [c (x — a) -f e (y — b) + gzf — h 2 z 2 — 0 , 
where a, (3. c, e, g, h are fixed constants ; a, b are the arbitrary parameters. 
(A.) The Discriminant. 
It is the same as that of the equation 
<*X 3 + /SY 3 + 3Z (cX + eY + gzZf - /tVZ 3 = 0 . 
Hence, 
S = — afice (g 2 + /r) z 3 , 
T = or/3 3 {2>g 2 — Id) 2 + 4a/3y (ae 3 4- /Sc 3 ) (3 Id — g' 2 ) z 3 — 4 h 2 (ote 3 — (3 o 3 ) 3 z 2 . 
Therefore, 
~{(oc/3 2 ) (3 g 2 - h 2 ) 2 z 3 + 4ra(3g (ae 3 + /Sc 3 ) (3/r - f)z - 4/P (ae 3 - /Sc 3 ) 3 ) 2 
^ — 64a 3 /3 3 c 3 e 3 (p 2 + /i 2 ) 3 z 3 
In order to show the way in which the factor z 4 arises, the method in which the 
discriminant is formed will now be examined. 
It may be obtained by eliminating X, Y, Z from 
(cX + eY + gzZ) (cX + eY + 3 gzZ) - h 2 z 2 Z 2 = 0 ... (133), 
aX 3 + 2 cZ (cX + eY + gzZ) = 0 
/ 8 Y 2 + 2 eZ (cX + eY + r/zZ) = 0 
Hence, 
Y = ± X /(ea/c/3). 
(134) , 
(135) . 
