242 PROFESSOR M. J. M. HILL ON THE LOCUS OF SINGULAR POINTS 
(D.) It wili be verified that the mean of the values of the parameter b, which 
correspond to the two surfaces having conic nodes, at a point on the locus £ = 0, 
is the same as the value of the parameter b, which is used to form the discriminant. 
The values of the parameters corresponding to the conic node are given by 
£ = 0, Bx (x — a) + ey (y — b) = 0, 
a (x — a) 2 -4- 2/3 (x — a) (y — b) -f- y (y — b) 2 + 2 g {x — a) -f- 2 h («/ — &) + nk(, a ~ 1 = 0. 
Hence 
(y — b ) 3 (ae 3 y 3 — 2/3Bexy + yBrx 2 ) -f 2 (y — b) Bx (hBx — cjey) -f nkB 2 x 2 £ > ‘ l ~ 1 = 0. 
Hence the mean of the values of y — b is 
Bx {gey — hBx)/(a.e 2 y 2 — 2/3Bexy -j- yB~x 2 ). 
Now putting £ = 0 in the value of y — b, given above in (A), the same result is 
obtained. 
(E.) This example is a case in which the assumption equivalent to that of Art. 7, 
viz., that 
D/ Ba D/ 35 
Da Bz 1)5 Bz 
at points on the locus of ultimate intersections cannot be made. 
The equations Df/Da = 0, Df/Db = 0 are given in (A). 
Hence Ba/Bz,. Bb/Bz are given by 
(«£ + B 2 x 2 ) ^ + (PC + Sexy) ^ = a (x - a) + £ (y - b) + g, 
(/3£ + Bexy) + (y£ + e 3 /) ^ = ft {x — a) + y {y — b) + h. 
Denoting for brevity 
a (x — a) + /3 {y — b) + g by G, 
f3 (x — a) + y (y — b) + h by H, 
£ 3 (ay — /3 3 ) + £ (ae 3 ?/ 3 — 2/3Bexy + yS% 3 ) by K, 
Bx {x — a) ey {y — b) by L, 
it follows that 
0“ = | [G (yi + «V) -H(M + Sexy)l 
| = |[H (a£ + 8%*) -G(K+ Se-n,)]. 
