390] SAME TWO LINES AND PASS THROUGH THE SAME FOUR POINTS. 
39 
It may be remarked that the equation of the conic passing through the three 
points and touching the axis of x in the point x = a. is 
(a-a) 2 (a-a") b (a! - a) 2 (a' - a) b' (a" - a) 2 (a - a') b" _ 
bx + ay-ab b'x + a'y -a'b' + b"x + d'y - a"b" ~ ’ 
and when this meets the axis of y we have 
^ (a- a) 2 (a' - a") p (a - a) 2 (a" -a) ^ (a" - a) 2 (a - a') 
to to Co ~ 
1 — 1 = o. 
y-b y-b y-b 
Hence, if this touches the axis of y in the point y = /3, the left-hand side must be 
- (a — a) 2 (a' — a") + —, (a' — a) 2 (a" — a) +^ T/ (a" — a) 2 (a — a') (y — /3) 2 
CL CL CL 
(y-b) (y-b')(y-b") 
and equating the coefficients of - , we have 
- (a — a) 2 (a' — a") + —, (a' — a) 2 (a" — a) + ^ (a" — a) 2 (a — a') 
CL CL 
- (a — a) 2 (a' — a") 4- —, (a' — a) 2 (a" — a) + (a" — a) 2 (a — a') 
CL CL CL 
(b + b' + b"-2/3), 
or what is the same thing, 
6 (6 ' + 6 "> („ _ a y (o' - o") + 6,(6 ", + i>) (o' - <*) ! (a" - a) + („" _ a y ( a - «') 
a a a 
= 2/3 [ - (a - a) 2 (a' - a") + (a - a) 2 (a" -a)+ h T , (a" - a) 2 (a - a') , 
1 a a a 
which gives /3 in terms of a, that is /3 l5 /3 2 , /3 3 , /3 4 in terms of a 1# a. 2 , a 3 , a 4 respectively. 
Cambridge, 30 November, 1863.