126 Professor Baker, On a property of focal conies 



be both touched by each of two other sections ^, S — and if, taking 

 a fifth arbitrary section, co, of the quadric, we measure the angle 

 between the planes of two sections a, 13, which touch one another, 

 in the usual way, by considering the homography of these planes 

 in reoard to the tangent planes drawn from their line of intersection 

 to the section co — ^then, with proper sense of measurement, [a, /3] 

 denoting the angle between these planes, we have 



[a, P] + IP, y] + [y, 8] + [S, «] = 0. 



Now take one of the two quadric cones containing the sections 

 a, y, and regard this cone, and the section co, as fundamental; 

 speak of a, y as circular sections of this cone, of opposite systems 

 because each has two points common with the other and with a». 

 Then we have ChasJes's theorem that a variable tangent plane of 

 a quadric cone makes angles of constant sum with two planes of 

 circular section of the cone, of opposite systems. 



§ 6. The reciprocal theorem is that a generator of a quadric 

 cone makes angles of constant sum with two conjugate focal lines 

 of the cone, that is, considering the conic in which the plane of w 

 cuts the cone, and the quadrilateral formed by the common 

 tangents of this conic and oj, makes angles of constant sum with 

 the lines joining the vertex of the cone, to an opposite pair of 

 intersections of two of these common tangents (Chasles, loc. cit., 

 § 827, p. 528). Projecting on to an arbitrary plane we have the 

 theorem that if P be a variable point of one of two conies having 

 S, H as common foci, the Cayley separations PS, PH in regard to 

 the other conic have a constant sum. An elementary proof can 

 be given depending on the fact that if PS meet the other conic 

 in >Si, S2, and PH meet the other conic in Hj^, H^, then, with proper 

 notation, each of S^H.,, S^H^ passes through a fixed point of the 

 line SH. 



§ 7, This theorem for conies is a particular case of the following : 

 Two conies V, W, have both double contact with a conic U, and 

 also both have double contact with another conic K. From a 

 point P oi K a tangent PX is drawn to F, and also a tangent P Y 

 to W; then the Cayley separations PX, PY, taken in regard to U, 

 have a constant sum (or difference) as P varies on K. Two tangents 

 are possible from the point P to the conic F; but the separation PX 

 is the same for both. 



If F degenerate into the pair of tangents to U from a focus S 

 of U, and W into the pair of tangents to U from the conjugate 

 focus H, then the conic K, touching these four tangents, will be 

 confocal with U, and the tangents PX, PY will become the lines 

 PS, PH. Thus the theorem includes that of § 6. 



