128 Professor Baker, On a pro;perty of focal conies 



where the coordinates in IJ, P, Q are those of the point P. If this 

 point be on the quadric K ^ 0, but not on U = 0, we thus get 



cos 6 cos <f) + a ^ ± sin ^ sin (f> 



showing that 9 ± (f) ^ constant, as was stated. 



§ 9. Now suppose a skew quadrilateral ABCD of which the 

 sides AB, BG both touch the quadric V, say in X and Y, respec- 

 tively, while the sides CD, DA both touch the quadric W, say in 

 Z and T respectively. The quadrics 7, W are supposed to have 

 two points of contact, so that quadrics can be drawn having ring 

 contact with both. Let TJ be one such; let K be another such 

 passing through C, and let A be on K. Then, considering Cayley 

 separations in regard to V, we have {BX), {BY) equal because F 

 has ring contact with U, and also {DZ), (DT) equal because W 

 ■ has ring contact with U. By § 8 we also have (AX) — (AT) equal 

 to (GY) — {GZ), if a proper sense be assigned to the separations 

 involved. 



We infer therefore that 



(AB) - (AD) = (AX) + (XB) - [(AT) + (TD)] = (AX) - (AT) 



+ (XB) - (TD) = (GY) - (GZ) + e{YB)-l (ZD), 



where e, ^ are each ± 1. Without making the proper detailed 

 examination, we shall put both e and ^ equal to 1, so obtaining 



(AB)- {AD) = {GB)- (GD). 



This is verified (§ 4) in the particular case where the quadrics F, W 

 coincide, there being then no need for the condition that A, G 

 lie on the same quadric K having ring contact with F and W. 



§ 10. A line joining a point of one focal conic to a point of 

 another focal conic of a confocal system of quadrics is a particular 

 case of a line touching two confocals of the system. And such a 

 line is part of a continuous curve which on either of these two 

 confocals may consist partly of arcs of the line of curvature which 

 is the intersection of these two fundamental confocals, and partly 

 of arcs of geodesies touching this line of curvature. As was recog- 

 nised by Chasles this continuous curve has everywhere the geo- 

 metrical property that if we take two other confocals of the 

 system, the homography of the tangent planes drawn to one of 

 these from a tangent line of the curve, in respect of the tangent 

 planes drawn to the other, is the same for every point of the curv^e. 

 That the analytic formulation may equally be regarded as uniform 

 for all parts of the curve seems often to be unnoticed; it is recog- 

 nised however by Staude in the papers above referred to. Let us 

 consider the system of confocals 



