and of bicircular quartics 123 



again respectively in P', Q', R', S', then (1) the points P', Q', R', S' 

 are equally on a plane, (2) if by the angle between the sections of 

 the quadric by the planes a, ^ be understood the Cayley separation 

 of these planes, measured by the homography of these planes in 

 regard to the two tangent planes to the quadric drawn from their 

 line of intersection, then the sum of the angles at P, R, determined 

 respectively by the sections a, ^ and y, 8, is equal to the sum of 

 the angles at Q, S, determined respectively by the sections /3, y 

 and 8, a. 



That P', Q', R', S' lie in a plane follows from the fact that the 

 four quadrics consisting of (i) the original quadric, (ii) the planes a, y, 

 (iii) the planes ^, 8, (iv) the planes PQR, P'Q'R', have seven, and 

 therefore eight points in common. For the relation between the 

 angles, denote by 6 the section by the plane PQRS, and in general 

 by (a, B) the angle between the sections (a, ^). Then we have 



77 ^ (d, a) + (a, ^) + (^, d) = {d, y) + (y, 8) -^ (8, d), 



and therefore 



(a, /3) + (y, 8) = 277 - {6, a) - (9, ^) - {d, y) - {d, 8), 



which is also the value of (^, y) + (8, a), the ambiguities of inter 

 pretation being properly settled in each case. 



In a plane we have the theorem that if P, Q, R, S be concyclic 

 points through which pass pairs of four circles a, ^, y, 8, namely 

 a, B through P, /?, y through Q, y, 8 through R and 8, a through S, 

 then the two angles (a, /3), (y, 8) have the same sum as the two 

 angles (^, y), (8, a); and this, not depending on the Axiom of 

 parallels, may well be regarded as a fundamental theorem. Further 

 if P' be the other intersection of a and ^, etc., the points P', Q', R', S' 

 are concyclic. The connexion of this result with the theorem of the 

 angles is incidentally remarked by Prof. W. McF. Orr, Trans. Camb. 

 Phil. Soc, XVI, 1897, 95. 



§ 2. Regard the bicircular quartic in question as the projection 

 on to an arbitrary plane of the section of a quadric by a quadric 

 cone of general position, the centre of projection being an arbitrary 

 point of the quadric. An arbitrary tangent plane of the cone cuts 

 the quadric in a section projecting into a conic having two points 

 of contact with the bicircular quartic, and this conic, passing 

 through the nodes of the quartic, is for us a bitangent circle, of 

 one mode of generation. The other three modes are obtained by 

 considering the other three quadric cones through the intersection 

 of the quadric and the first cone. Take then two bitangent 

 circles of the bicircular quartic of the first mode of generation, say 

 a and y; their points of contact will be on another circle, say/?, 

 as appears from the three dimensional figure. Take also two 



