46 
ON THE CONICS WHICH PASS THROUGH TWO GIVEN POINTS 
[392 
the equation is 
where 
YQ + X 2 = 0, 
X = x — y, 
Y = x — y V (run), 
2 
Q = fr( m )-y/(njp ^ 1 +V ( mn )l 0* “ 2/) + i 1 “ V ( mn )J z ] ■ 
these values give 
sc-y =X, 
x — y V (tow) = F, 
{1 — V (tow)} z = {V (to) — V (w)} 2 Q + 2 {1 + V (tow)} X, 
or, what is the same thing, 
{1 — V (run)) x = — V (tow) X + Y, 
(1 — V (tow)} y = — X + Y, 
{1 — V (tow)} z = 2 (1 + V (tow)} X + (V (to) — V (w)} 2 Q, 
whence also 
(1 - V (tow)} (x + y + z) = {1 + V (mn)} X+2 F+ {V (m) - V (w)} 2 Q, 
or the equation of the line infinity is 
{1 + V (tow)} X + 2 F + {V (to) - V (»)}■ Q = 0, 
a formula which may be applied to finding the asymptotes and thence the centre of 
the conic 
YQ + X 2 = 0. 
In fact we have identically 
[2kx + 2ky — (2k + 1) z} 2 — (1 + 4k) (2kx — z) 2 = 4& 2 (x + y + z) 2 — Yk (1 + 4&) (kx 2 + yz), 
that is 
- 4tk (1 + 4&) (kx 2 + yz) = {2kx + 2ky - (2k + 1) z} 2 - (1 + 4>k) (2kx - z) 2 - 4k 2 (x + y + z) 2 , 
which, if x + y + z = 0 is the equation of the line infinity, puts in evidence the 
asymptotes of the conic kx 2 + yz = 0. Hence writing ax, /3y, yz in the place of x, y, z 
ItQ? /3 
respectively, and ' = k', that is, k = k', we have 
- 4 | 7 Id (l + *fh. k) (Kx? + yz) = Kx + 2 ^ 7 Ky - (2 l3 J K + \yz^ 
(*§* k’x - yzj - 4 ffl!* * (ax + l 3y + yzf,