44 
ON THE CONICS WHICH PASS THROUGH TWO GIVEN POINTS 
[392 
To fix the ideas I take m and n each positive and mn > 1 ; also I attend first 
to the series where \f {mn) is taken positively. At the points where the conic meets 
infinity, we have 
{V (m) + V («)} V {xy) = x + y V {mn) - 7 {x + y), 
which gives two coincident points, that is the conic is a parabola, if 
(1 - 7) {V {mn) - 7} = i {V («0 + V 0)} 2 > 
that is 
r - 7 {1 + V («»)} = 4 {V (m) - V ( n )Y> 
or 
7 = 2 [1 + V («*») ± V {(1 + m) (1 4- n)}], 
where it is to be noticed that 
7 = i [1 + V ( mn ) + V {(1 + m) (1 + n)}] 
is a positive quantity greater than V {mn), say 7 = p, 
7 = i [1 + V 0*0 - V {(1 + m) (1 + n)}] 
is a negative quantity, say 7 = — q, q being positive. 
The order of the lines is as shown in fig. 1, see plate facing p. 52. 
7 = — 00 to 7 = —q, curve is ellipse; 7 = — q, parabola P 2 , 
7 — — q to p, curve is hyperbola; 7 =p, parabola P lf 
7 =p to 7 = 00 , ellipse. 
Resuming the equation 
{x — my) {x - ny) + 2 [x + y V {mn)) yz + 7-z 1 = 0, 
the coefficients are 
{a, b, c, f, g, h) = {1, mn, y 2 , 7 V {mn), 7, -£(ra + w)}, 
and thence the inverse coefficients are 
{A, B, G, F, G, H) = 
[0, 0, - £ (w - w) 2 , - i 7 {v (m) + v O)} 2 , - 17 v («m) {V (m) + V (w)} 2 , 
K = - Ì7 2 w O) + V (*0} 4 > 
or, omitting a factor, the inverse coefficients are 
{A, B, G, F, G, H) = 
°» °> 2^ {V O) 
V (*01 2 > 1, V W, - 7 
Considering the line 
\x + y,y + vz = 0, 
the coordinates of the pole of this line are 
x : y : z— — 7/U. + V {mn) v 
: — <y\ + v 
W {VO) + \/(>0} 2 ]- 
: V {mn) A + p + 
_1 
2 7 
W O) - V 00} 2 v,