146 
PROFESSOR M. J. M. HILL ON THE LOCUS OF SINGULAR POINTS 
P/(£q, g, «) 
Y)rj 
ty (-??, £«) 
~ ur 
= o 
= 0 
• • ( 21 ), 
, . ( 22 ), 
and the equations obtained from (19)-(22) by changing i, 77, £, a into £ + Sf, 77 + 817 , 
£ -f- S£, a + 8a respectively. 
Denoting differential coefficients by brackets containing the independent variables, 
with regard to which the differentiations are performed, these last equations become 
by means of (19)-(22) 
[a] (Sa) = 0 .( 23 ), 
By (23), 
[£ m + [£ V ] (817) + [£ £] (8£) + [£ «] (Sa) = 0 
lv> £] (8f) + b, T?] (817) + [ 17 , £] (S£) + [77, a] (8a) = 0 
[£> £] (8f) + [£, 77] (877) + [£, £] (S£) + [£, a] (Sa) = 0 
t] = 0 
• (24), 
• ( 25 ), 
• ( 26 ). 
at every point of the conic node locus. Hence the co-ordinates of every point on the 
conic node locus satisfy the equation of the locus of ultimate intersections.* 
Further, since [a] = 0 at every point on the conic node locus, the corresponding 
equation is satisfied at the conic node on the surface (18). 
Hence 
[a, (Si) + [a, 77] (877) + [a, £] (S£) + [a, a] (Sa) = 0 . . . ( 27 ). 
Since equations ( 24 )-( 27 ) must give consistent values for S£ : 877 : S£ : Sa, it follows 
that the Jacobian 
Dj ffiH. [awi _ 0 
■ (28). 
* I am indebted to Dr. Forsyth for the following example:— 
Let the surfaces be 
(cc — a ) 2 + (y — a) 2 — /r 2 (z — af = 0. 
The discriminant is 
- 2 (2* _ * _ y y + (,2 _ 2 ) (a - yf. 
Hence the locus of ultimate intersections consists of the two planes 
k(2z — «—*?/) = ± ^2 — k~(x — y ). 
These planes intersect in the straight line * = y = 0 , which is the locus of conical poiuts of the 
surfaces. 
