and of bicircular quartics 125 



theorem in regard to the angles, if we consider the homography 

 of A,B in regard to the quadric, say a, and take the corresponding 

 homographies for the pairs B, G; C, D; D,A respectively, say b, c, d, 

 we have ac = bd, or a/d = b/c. In words, the difference of the 

 Cayley separations of A from 5 and D, in regard to the quadric, 

 is the same, for unaltered positions of B, D on the second conic, 

 when A is replaced by any other point C of the first conic. 



This result includes the particular case of the focal conies of 

 a confocal system, for which we may also consider the further 

 particular case of actual Euclidian distances between the points. 

 (Cf. § 10 below, where the relation between the separation and the 

 distance is given.) 



§ 4. If we assume that the sides of the skew quadrilateral 

 ABCD in § 3 touch an enveloping cone of the quadric, we can 

 deduce the relation between the Cayley separations in another 

 way. In fact if the sides of a skew quadrilateral touch any quadric 

 having ring contact with a given quadric, the sum of the Cayley 

 separations belonging to the sides of the quadrilateral, each taken 

 in proper sense, is zero, the separations being measured by the 

 latter quadric. For if ^T be a tangent to a quadric V, which has 

 ring contact with a quadric U, drawn from a point A, the Cayley 

 separation AT in regard to U is independent of T. If A be 

 (^, 77, ^, r), T be (x, y, z, t), so that, with usual notation, V^. = 0, 

 7^^ = 0, and Z7 be 7+ P^^ 0, then TJ^ = PJ^, U^^ = PJ"^, and hence 



which is independent of x, y, z, t; and U^^KUJJ^Y is the cosine of 

 the separation in question. Therefore, if the ^iAqsAB, BC, CD, DA 

 of the skew quadrilateral touch 7 respectively at L, M, X, Y, 

 we have the following relations among the separations 



(AB) = (AL) - (BL), (BC) = (BM) - (CM), 



(CD) = {CX) - (DX), (DA) = {DY)- (AY), 



{AY) = iAL), {BL) = {BM), {CM) = {CX), {DX) = {DY), 



leading to 



{AB) + {BC) + {CD) + {DA) = 0, 



or {AB) - {AD) = {CB) - {CD). 



In the application of this result above, 7 was a cone. 



§ 5. We may however make an application in which C/ is a 

 cone, and 7 not a cone, U being an enveloping cone of 7. Namely, 

 if the sides of a skew quadrilateral touch a quadric, the sum of 

 the four Cayley separations of the vertices, each in proper sense, 

 in regard to any enveloping cone of the quadric, is zero. The 

 reciprocal theorem, is that if two plane sections a, y of a quadric 



