Professor Baker, On the Hart circle of a spherical triangle 119 



upon this inscribed circle, the point of contact with the Hart circle, 

 which touches this and certain other three tangent circles of the 

 sides of the triangle, is the apolar complement of I, J in regard to 

 P, Q, R. For the particular case of the nine point circle of a plane 

 triangle the result has been remarked by Prof. F. Morley, as was 

 pointed out to the writer by Mr J. H. Grace, Bulletin of the American 

 Math. Soc, I, 1895, 116-124 ("Apolar triangles on a conic"). 



§ 4. Another result may also be stated here. To introduce it 

 and render its meaning clearer we state it first for the Hart circle 

 of a spherical triangle in Euclidian geometry. If this circle meet 

 the sides of the spherical triangle ABC respectively in U, U' on BC, 

 V, V on CA, W, W on AB, then, with proper choice of notation, 

 the arcs AU, BV, CW are concurrent, say in S, and the arcs 

 A'U', B'V, CW are concurrent, say in S' . The result in question 

 is that S, S' are the centres of similitude of the circumscribed 

 circle of the triangle ABC and the Hart circle. It is a direct 

 generalisation of the corresponding familiar fact for the nine point 

 circle of a plane triangle. 



Stated in the more general way here adopted, which is also the 

 more precise way, the theorem is that the lines OS, OS' are each 

 the intersection of two planes through which touch both the 

 section m and the section by the plane ABC. If PQR, P'Q'R' be 

 two sets of three points lying respectively on two plane sections 

 of a quadric, such that PP' , QQ' , RR' are concurrent, the sections 

 lie on a quadric cone having this point of concurrence for vertex; 

 thus a plane through touching the section [x by the plane ABC 

 equally touches the section /x' by the plane A'B'C . Now S, the 

 point of concurrence of AU, BV, CW, is the vertex of one cone 

 containing the sections //,, xd; and S' is similarly the vertex of one 

 cone containing the sections jx' , rn. The line OS', joining the vertex 

 of one cone containing the sections fx', m to the vertex of one cone 

 containing the sections fj., fj,', passes through one of the vertices 

 of the two cones containing the sections, /jl, m; as OSS' are not 

 collinear, this line OS' passes through the vertex other than S of 

 a cone containing /x and w. The two cones containing fx and m 

 thus have their vertices on OS and OS'. Now to each of these 

 cones there can be drawn from two tangent planes, which 

 intersect in the line joining to the vertex of the cone; the four 

 planes so obtained touch the sections fi and w, and thus are the 

 four common tangent planes of the cones with vertex standing 

 on the sections /x, ta. Two of these planes therefore intersect in 

 OS and two in OS'; which is the result we desired to obtain. 



There are as we have said eight sections m each touched by four 

 of the common tangent planes of the sections in YOZ, ZOX, XOY. 

 These tall into four pairs, the planes of a pair intersecting on the 



