PROFESSOR CAYLEY OF POLYZOMAL CURVES. 35 



imaginary points (A 2 , B 2 , C 2 , D 2 ) are only given as the intersections of an imagin- 

 ary circle (centre real and radius a pure imaginary) by a pair of real lines. The 

 points (C 2 , A 2 ) qua anti-points of (C, A) are easily constructed as the intersections 

 of a real circle by a real line, and the like as to the points (B 2 , D 2 ) qua anti-points 

 of (B, D), but the construction for the two pairs of points cannot be effected by 

 means of the same real circle. 



Property in regard to Four Confocal Conies — Art. Nos. 78 to 80. 



78. All the conies which pass through the four concyclic points A,B,C, D, have 

 their axes in fixed directions; but three such conies are the line-pairs {BO, AD), 

 (CA, BD),and (AB, CD), whence the directions of the axes are those of the bisec- 

 tors of the angles formed by any one of these pairs of lines ; hence, in particular, 

 considering either axis of a conic through the four points, the lines AB and CD 

 are equally inclined on opposite sides to this axis, and this leads to th theorem 

 that the anti-points (A 3 ,B 3 ) (C 3 ,D 3 ) are in a conic confocal to the given conic 

 through (A, B, C, D); whence, also, considering any given conic whatever through 

 (A, B, C, D), the points (A x , B x , C 1% A)> K* B v °v A) ( A v B v °v A) lie seve- 

 rally in three conies, each of them confocal with the given conic. 



79. To prove this, consider any two confocal conies, say an ellipse and a hyper- 

 bola, and let F be one of their four intersections ; join F with the common centre 

 0, and let OT, ON be parallel to the tangent and normal respectively of the ellipse 

 at the point F. OF, OT are in direction conjugate axes of the ellipse, and OF, 

 ON are in direction conjugate axes of the hyperbola ; and if they are also the axes 

 in magnitude, that is, if the points T, iVare the intersections of T with the 

 ellipse and of ON with the hyperbola respectively, then it is easy to show that 

 OT 2 + ON 2 = 0. And this being so, imagine on the ellipse any two points A, B 

 such that the chord AB is parallel to OT, that is conjugate to OF; AB is bisected 

 by OF, say in a point K, or we have parallel to OT the semichords or ordinates 

 KA —KB ; and we may, perpendicularly to this or parallel to ON, draw through 

 K in the hyperbola a chord A 3 B 3 , which chord will be bisected in K, or we shall 

 have KA 3 = KB 3 . Hence KA, KA, are in the ellipse and the hyperbola respec- 

 tively ordinates conjugate to the same diameter OF, and the semi-diameters con- 

 jugate to OF being OT, ON respectively, we have KA 2 ( = KB 2 ) : KA]( = KBl) 



= OfiON 2 , that is, KA 2 = KB 2 = — KA\= —KB\\ or (A S ,B 3 ) will be the 

 anti-points of (A,B). 



80. Conversely, if in the ellipse we have the two points (A,B), then drawing 

 the diameter OF conjugate to AB, and through its extremity F, the confocal 

 hyperbola, then the anti-points (A 3 , B 3 ) will lie on the hyperbola. And similarly, 

 if on the ellipse we have the two points (C,D), then drawing the diameter 

 OCr conjugate to CD, and through its extremity G a confocal hyperbola, the 



