AND LINES IN THE INTERSECTIONS OF A SYSTEM OF SURFACES. 
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(B.) 1'he Locus of two Conic Nodes is 4 = 0, and if n = 2 there is cdso a Curve 
Locus of Conic Nodes. 
The singular points are determined by finding solutions of f — 0, BfjBx = 0, 
Df/Dy = 0, Bf/Dz = 0. 
But since D/’/D £ = DfjDz, the equations 
/= 0, D f/Dx = 0, D//Dy = 0, D//D£ = 0 
may be used instead, where a;, y, £ are now the independent variables, so that the 
meaning of the symbol of differentiation D is changed. 
The equations to be satisfied are 
(«£ + W) (x — af + 2 (/3£ + Sea:y) (a: — a) (y — 8) 4- (y£ + e 2 y 2 ) (y ~ 
+ 2y£ (a; - a) 4- 2A£ (y — 6) + A£* = 0, 
[&» (a; — a) + ey (y — 6)] 8 (2a: — a) + £ [a (x — a) + /3 (y -- 8j + y] = 0, 
[Sa: (a: — o) + ey (y — 8)] e (2y — 8) + £ [/3 (a: — a) + y (y — h) + A] = 0, 
a(x — df +2/3 (a: — a) (y —- 8) + y (y — &) 2 4 2y (a 1 — a) + 2A (y — 8) + nkf l ~ l = 0. 
From these it follows that 
[8a: (a: — a) + ey (y — 8)] 2 -f 8(I — «) £ f< = 0. 
(i.) One method of solving the above equations is to take 
£ = o, 
8a: (x — a) + ey (y — 8) = 0, 
a (a: -- a) 2 + 2/3 (a — a) (y — 8) -f y (y — 8) 2 4- 2y (a: — a) 4- 2A (y — 8) 4* ^£"“ 1 s= 0. 
Hence whether n— 1 or 2, there are two values of 8, and tw*o corresponding values 
of a. Hence there are two conic nodes. Hence £ — 0 is a locus of two conic nodes, 
(ii.) Another method of solving the equations is to take 
8 (2x — a) _ u (x ~ a) 4 /8 (y b) + g 
e (2 y -b) ~~ /3 (a - a) + y {y -b) + li' 
[Sa: (i x — a) + ey (y — 8)] 8 (2a: — a) 4- £ [a (a — a) 4' (3 (y — 8) 4- g] — 0, 
a(x — af 4 2/8 (x — a) (y — 8) 4- 7 (y ~ fy 2 + 2 g (x — a) 4- 2 h (y — 8) 4- nk£ n ~ x = 0, 
[8a- (a: - «) f t y (y - 8)] 2 4- A (1 - ?i) £" = 0, 
