17G BOOK OF MATHEMATICAL PROBLEMS. 



its centre at one of the four points, and the other three self- 

 conjugate. These conies are therefore similar and similarly 

 situated. Moreover, if we draw the two parabolas which can 

 pass through the four points, their axes must be parallel respect- 

 ively to coincident conjugate diameters of any one of the four 

 conies ; i. e. to the asymptotes. But the axes of these para- 

 bolas must be parallel to the asymptotes of the couic which is 

 the locus of centres of all conies through the four points ; since 

 when the centre is at infinity the conic becomes a parabola. 

 Hence, finally, if we have four points in a plane, the four conies, 

 each of which has its centre at one of the four points, and the 

 other three self-conjugate, and the conic which is the locus of 

 centres of conies through the four points, are all similar and 

 similarly situated. 



Let A, B be any two fixed points on a' circle, oo, oo' the two 

 impossible circular points at infinity, P any other point on the 

 circle; then P{A oo oo'J5} is constant. Hence PA, PB and the 

 circular points divide the line at infinity in a constant anhar- 

 monic ratio. Hence, two straight lines including a constant 

 angle, may be projected into two straight lines, which divide 

 the straight line joining two given points (the projections of the 

 circular points) in a constant anharmonic ratio. In particular, if 

 APE be a right angle, AB passes through the centre (the pole 

 of oo oo'), and the ratio becomes harmonic. 



Thus, projecting the property of the director circle of a conic, 

 we obtain the following theorem : " The locus of the intersection 

 of tangents to a conic which divide harmonically the straight line 

 joining two given points is a conic passing through the given 

 points; and the straight line joining the two points has the same 

 pole for both conies." 



If tangents be drawn to any conic through the circular points, 

 their four points of intersection are the real and impossible foci of 

 the conic. If the conic be a parabola, the line joining the cir- 

 cular points is a tangent, and one of the real foci is at infinity, 

 while the two impossible foci are the circular points. Many 



