238 PROFESSOR M. J. M. HILL ON THE LOCUS OF SINGULAR POINTS 
If' n = 1, then Sx (x -- a) -J~ ey (y — b) = 0, 
Hence £ = 0, and this is the same solution as in (i.). 
If, however, n — 2, there are four equations to be satisfied by x, y, £. Eliminating 
x, y, £ it is necessary that a certain relation should be satisfied by a, b, in order that 
the equations may be consistent. 
From the above equations (when n = 2) 
§x (x —■ a) -j- ay (y — b) = \/ k £ 
v / k S (2 x — a) 4 a (x — a) 4 /3 (y ~ b) 4 g = 0, 
rejecting a solution £ = 0. 
v / k e (2y — b) -f A (a? — a) 4 y (y — 6) + h — 0, 
a(x — af + 2 ft(x — a) {y — 6) 4 y (y — 6) 2 + 2g (cc — a) 4 2/? (y — 6) 
4 2 v/k [Sas (& — a) 4 ey (y — Z/)] = 0. 
Hence 
0 (a* — a) + A (*/ — 6) + v 7 * [Sa (a; - - n) 4 eb (y — 6)] = 0. 
Hence 
(x — a) (a 4 2S\/k) 4 (y — b) ft 4 (g 4 ah\/ k) = 0, 
~ a ) ft (y ~ b) (y ~ f- 2e\/ k) 4 (A 4 b&\/ k) — 0, 
(« — a) (</ 4 aS\/*) 4 (y Z>) (A 4 bey/ k) = 0. 
Hence 
a 4 2 S\/k /S 
£ y 4 2e\/ k 
g 4 ctS \/k h 4 be\/ k 
i.e., 
(g 4 a S\/k) 2 (y 4 2ev/#c) —• 2/3 4 a Sv 7 k) (li 4 bey/, k) 
4 (h 4 be\/ k)~ (a 4 2 Sv/ k) = 0. 
Hence only when this relation holds between a, b , will there be any conic node on 
the surface which is not also on the locus £ = 0. 
As in the general theory explained in Art. 3, this leads to a curve locus of conic 
nodes. It need not therefore be further considered.’ 
Hence the only locus of conic nodes that need be considered in the discussion of the 
discriminant is £ = 0. 
Now, whether n = 1 or 2, the lowest power of £ in the discriminant is £ 2 ; hence 
this factor is accounted for. 
g 4 oSv/ k 
h 4 be \/k 
0 
= 0 , 
