120 Professor Baker, On the Hart circle of a spherical triangle 



polar plane of 0, being harmonic conjugates in regard to this plane 

 and 0; for the pair associated as above with the two planes ABC, 

 A'B'C the lines OS, OS' are the same. There is another pair 

 associated similarly with the planes AB'C , A'BC, a third pair 

 with the planes BC'A', B'CA and a fourth pair associated with 

 the planes CA'B', CAB. And it may be remarked that the sections 

 by the planes ABC, AB'C, BC'A', CA'B' are all touched by four 

 planes, as follows at once from the fact that AA', BB', CC are 

 concurrent; so also the sections by the planes A'B'C, A'BC, B'CA, 

 CAB are all touched by four planes. 



§ 5. Another remark may be made, relating to a property which 

 appears in Euclidian geometry as Salmon's theorem that the 

 tangent of the radius of the circumcircle of a spherical triangle is 

 twice the tangent of the radius of the Hart circle. 



Let P be the pole of any plane section of a quadric, upon which 

 any point A is taken, and be any other point; denote by p the 

 Cayley separation of the lines OP, OA in regard to the quadric, 

 and by S the Cayley separation of P from 0. It can then be shown 

 that p is independent of the position of A upon the section, and is 

 indeed symmetrical in regard to P and 0, being connected with 8 

 by an equation sin S sin p = ± 1. Calling p the radius of the section 

 in regard to the point 0, it can be shown that if p, p' be the radii 

 of any two sections a, a' whose planes intersect in a line I, and the 

 planes joining I to and to the vertex of one of the two cones 

 containing a and a' be respectively denoted by co and y, then 

 tanp/tanp' is equal to the homography (y, a>; a, a') or to the 

 negative of this. In particular when the planes y, a are harmonically 

 separated by to and a', this leads to tanp = 2 tan p'. In our figure 

 the plane a, which is the polar of S in regard to the quadric, passes 

 through the line of intersection of the planes ABC and w, since S 

 is the vertex of one of the cones containing the section by ABC 

 and the Hart section zu, and this plane a also contains the vertex 

 of the other cone containing these sections; it can easily be proved 

 that the plane co which joins to the line of intersection of the 'planes 

 ABC and m is harmonically separated from m by the planes ABC 

 and a; thus the planes a, ABC, oj, w have the relation of the 

 respective planes y, a, w, a in the general description just given. 

 It follows tbafc if p, R be the radii of the sections m, ABC, we have 

 tan -R = 2 tan p ; which is what we wished to prove. 



§ 6. A last remark may be added bringing into relief the con- 

 nexion between the present point of view and that of the Euclidian 

 geometry. As hitherto, let OXYZ be a tetrahedron whose faces 

 meet a quadric in sections having four common tangent planes. 

 Denote by ia, i^, iy the Cayley separations OX, OY, OZ in regard 



