36 THEOREM RELATING TO THE FOUR CONICS WHICH TOUCH THE [390 
or, assuming as we may do, K = — (af — a") (a" — a) (a — a), this gives 
L a =(a — a) 2 (a — a"), 
L a' = (a' — a) 2 (a" — a ), 
L"a" = (a" -of (a -a' ). 
But in the same manner, if the conic touch the axis of y, say at the point y = /3, 
we have 
Lb =(b -/3) 2 (6' -6"), 
Lb’ = (6'-/3) 2 (6"-6 ), 
L”b" = (b" — /3) 2 (b -6'); 
and thence 
b (a —a) 2 (a' — a") : b' (a — a) 2 (a" — a) : b" (a" — a) 2 (a — a) 
= a(b —13) 2 (b’ - b”) : a’ (b' - /3) 2 (b" - b) : a" (6" - /3) 2 (6 - b’). 
Putting 
P = a b (a' - a") (6' - b"), 
P' = a'b' (a”-a )(b"-b ), 
P” = a"b" (a — a' )(b -b’\ 
we have 
(a-*Y | : («'-«)»- : (a" - af^ = (b- -b'J : (6'-£)*(&"-6)’ : (b" - f3f (b - bj; 
cr it it 
and thence 
(a 
- a) : (a' - a) : (a" - a) 
which gives 
= (6-/3) (6'-6") : (6'-/3)(6"-6) : (6" -/3) (6 - 6'), 
(a-.)^) + ((l '- a )^-P + ( a "- a )Vip) = 0, 
x ' a 'a v 'a 
and we have in like manner 
(6 - /3) + (6' - /3) + (6" - /3) v = 0, 
but the first of these equations is alone required for the present purpose, 
for shortness 
P = a 2 X, P’ = a' 2 X', P" = a" 2 X”, 
Putting 
the equation is 
(a - a) V (X) + (a’ - a) V (X') + (a" - a) V (X"),