AND TOUCH TWO GIVEN LINES. 
4 7 
392] 
that is 
- 4/3 2 7 2 /c' (a 2 + 4/3yV) (k'x 2 + yz) = {2a/3yk'x + 2/3 2 yk'y - (2¡3yk' + a 2 ) yz} 2 
— (a 2 + ^¡3yk') (2/3yk'x — ayz) 2 - 4</3 2 y 2 k' 2 (ax + /3y + yz) 2 , 
or, what is the same thing, 
- 4/3 2 &' (a 2 + 4<f3yk') (k'x 2 + yz) = (2a/3k'x + 2/3 2 k'y - (2/3yJc + a 2 ) z} 2 
— (a 2 -f 4/3yk') (2/3k'x — azf — 4/3 2 k' 2 (ax + ¡3y + yz) 2 , 
which, when ax + /3y + yz = 0 is the equation of the line infinity, puts in evidence the 
asymptotes of the conic k'x 2 + yz = 0. 
Now writing X, Y, Q in the place of x, y, z\ k’— 1, and a = {1 + V (mn)}, /3 = 2, 
Y = {\/ (m) — V (n)} 2 , we have 
- 16 [{1 + V (mn)) 2 + 8 {V (m) - V(n)) 2 ] (YQ + X 2 ) 
= [4 {1 + V (mn)) X + 87 - (4 {V (m) - a/ (re)} 2 + (1 + V (mn)) 2 ) Q} 2 
- [{1 + V (mn)) 2 + 8 {V (m) - V (n)} 2 ] [4X - {1 + V (mn)) Q] 2 
— 16 [{1 + V(mn)) X + 27+ (V(m) — V (n)} 2 Q] 2 , 
and the asymptotes are 
4 {1 + V (mn)} X + 8 7 - [4 {V (m) - V (n)} 2 + {1 + \f (win)} 2 ] Q 
= + V{1 + V (mn)) 2 + 8 (to) — V (n)) 2 [4X — {1 4- V (win)} Q]. 
At the centre 
4 {1 + V (mn)) X + 8 7 - [4 {V (to) - V (n)} 2 + {1 + V (mn)} 2 } Q = 0, 
4X — {1 + V (mn)} Q = 0, 
but the first equation is 
{1 + V (mn)} [4X - Q {1 + V(mn)}} + 87- 4 {V (ni) - V (n)} 2 Q = 0, 
so that we have 
4X = {1 + V (win)} Q, 2 Y = (to) — V 0^)}“ Q> 
the first of these is 
2 {V (to) - V (n)} 2 (x - y) - {1+ \f (mn)} 2 (x-y)-(l- mn) z = 0, 
and the two together give 
2X {V (to) - V (n)} 2 - {1 + V (mn)} 7=0, 
so that we have 
2 {V (to) - V (n)} 2 (x - y) - {1 + V (mn)} [x-y \/ (mn)} = 0, 
[{1 + V (win)} 2 - 2 {V (to) - \/ (n)} 2 ] (x - y) + (1 - win) ^ = 0, 
to determine the coordinates of the centre.