48 ON THE CONICS WHICH PASS THROUGH TWO GIVEN POINTS [392 
The equation of the chord of contact is 
x + y V (ran) + yz = 0, 
which for 7=1 is parallel to y — 0 and for 7 = V (ran) is parallel to x — 0. But the 
coordinates of the centre are 
x : y : 2 = — 7 + V (mn) : - 7 + 1 : V (mn) + 1 + - 1 - {V (m) - V (n)} 2 , 
which for 7=1 give 
y = 0, x : 2 = — 1+V (mn) : V (mn) + 1 + \ {V (m) — V (n)} 2 , = — 2 + 2 ^ (mn) : 2 + m + n, 
and for 7 = V (mn) give 
x = 0, 
I 7Y) -1- 11 
y : 2 = 1 - V (mn) : V (mn) + 1 + ^ W ( m ) ~ V ( w )} 2 , = 2 ~ 2 \/(mn) : 2 \/(wn) + • 
The line drawn from the fixed point on the chord of contact to the centre has for 
its equation 
x + y V (ran) + y z = 0, 
where, writing for a?, y, z the coordinates of the centre, we have 
- 7 {1 + V (ran)} + 2 \/ (ran) + 7' 
V (mn) + 1 + ^ (V (w) - V (n)} 2 
= 0, 
that is 
, _ 7 11 + V (mn)} — 2 V (mn) 
7 = 
1 + V (ran) + ^ (V (ra) - V (n)} 2 
or, what is the same thing, 
7 7 {V (ra) - V (n)} 2 + 27 {1 + V (ran)} ’ 
2 7 
- 7 IV (ra) + V (n)} 
and consequently y = 7 only for 7 = 0. 
It is now easy to trace the corresponding positions of the chord of contact through 
the fixed point [x + y V (ran) = 0, z = 0}, and of the centre on the hyperbola which is 
the curve of centres: see fig. 2 in the plate facing p. 52. 
The lines 0P. 2 , OL, 0©, 0P 1} OX, OG, OH are positions of the chord of contact, 
and the points P«, L, ©, P 1} X, G, H, on the hyperbola which is the curve of centres 
are the corresponding positions of the centre.