264 ON THE CONICS WHICH PASS THROUGH THREE GIVEN POINTS, &C. [342 
the axes as above, we avoid the consideration of the several cases corresponding to 
different signs of these quantities. And this being so, if x = a is the coordinate of 
the point of contact, the equation of the conic is 
(A, B, G, F, G, H\x, y, l)- = 0, 
where 
A = 2bb' (a' — a), 
B = 2 a- (a' — a), 
C = 2a 2 bb' (a' — a), 
F = — a? (a — a) (b' + b), 
G = — 2a.bb' (a' — a), 
H = {(a 2 + aaf) (b' — b) — 2a (ab' — a'b)}, 
and these give 
AB —R 1 = — [{(a — a) \J (b') + (a — a') (Z>)} 2 — (a' — a) (a'b — ab')] x 
[{(a — a) V (b') — (a — a') ^ (b)] 2 — (a' — a) (a'b — ab')], 
BG — F 2 = — a 4 (a' — a) 2 (b' — b) 2 , 
CA - G 2 = 0, 
GH — AF = — 2bb' (a' — a) (b' — b) a (a — a) (a — a'), 
HF-BG =- (a' - a) (V - b) a 2 \(b' + b) (a 2 + aa') - 2a (ab' + a'b)], 
FG - CH = — 2bb' (a' - a) (b' - b) a 2 (a - a) (a - a'). 
The condition that the conic may be a parabola is AB — H 2 = 0, which gives, as it 
should do, four real values of a. 
2, Stone Buildings, W.G.