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THEOREM RELATING TO THE FOUR CONICS WHICH TOUCH 
THE SAME TWO LINES AND PASS THROUGH THE SAME 
FOUR POINTS. 
[From the Quarterly Journal of Pure and Applied Mathematics, vol. vm. (1867), 
pp. 162—167.] 
The sides of the triangle formed by the given points meet one of the given lines 
in three points, say P, Q, R; and on this same line we have four points of contact, 
say A lf A.,, A 3 , A 4 ; any two pairs, say A 1 , A 2 ; A 3 , A 4 , form with a properly selected 
pair, say Q, P, out of the above-mentioned three points, an involution; and we have 
thus the three involutions 
(A, A 5 A 3 , A; Q> P)> 
(A> -A > -A, -A j P> P\ 
(A, A; A 2 , A; P, Q). 
To prove this, let x = 0, y = 0 be the equations of the given lines, and take for 
the equations of the sides of the triangle formed by the given points 
bx + ay — ab = 0, 
b' x +a' y —a' b' =0, 
b"x + a"y-a"b" = 0: 
the equation of any one of the four conics may be written 
Lab L'a'b' L"a"b" 
bx + ay — ab^ b'x + a'y — a'b' b"x + a"y — a'b" 
and if this touches the axis of x, say at the point x= a, then we must have 
La L'a L"a" - K (x - a)* 
x - a + x - a + x - a” ~ (x - a) (x - a') (x - a") ’ 
5—2