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



On the Hart circle of a spherical triangle. By Professor 

 H. F. Baker. 



[Read 9 February 1920]. 



This note is concerned with the problem, given three arbitrary 

 plane sections of any quadric, of finding a fourth section which 

 shall be tangent to four of the tangent planes of the three given 

 sections. If the three given sections are concurrent on the quadric 

 they have only four tangent sections, and the fourth section is 

 unique, the projection of the figure on to a plane (from the point 

 of concurrence) giving rise to Feuerbach's theorem of the nine- 

 point circle. In general the three given sections have eight common 

 tangent planes; in fact any two of these sections lie on two quadric 

 cones, and the six vertices of the cones so obtainable lie by threes 

 on four coplanar lines; the three cones whose vertices are on any 

 one of these lines have a pair of common tangent planes, which 

 thus touch the three sections. The eight tangent planes of these 

 are thus accounted for. There are now fourteen ways of selecting, 

 from these eight tangent planes, four which all touch another 

 section ; six of these ways, in which the four tangent planes selected 

 are tangent to a fourth section passing through the point of con- 

 currence of the three given sections, are easy to recognise, and do 

 not need further consideration. There are however eight ways of 

 choosing four from the tangent planes which shall all touch another 

 section lying in a plane w forming with the planes of the three given 

 sections a finite tetrahedron. 



§ 1. We are thus lead to the problem of the condition necessary 

 and sufficient in order that the sections of a quadric by the four 

 faces of a tetrahedron should have four common tangent planes; 

 and the main object of this note is to state this condition in a 

 form which in fact leads to great simplification of what is generally 

 presented as a somewhat intricate theory, and to point out several 

 results, apparently new, which follow from this. Let the tetra- 

 hedron be 0, X, Y, Z; denote the intersections of the quadric 

 with OX hy A, A', those with OY by B, B' and those with OZ 

 by C, C"; similarly denote the intersections with YZ by V, U', 

 those with ZX by F, 7' and those with XY by W, W. In general, 

 if each edge of the tetrahedron be joined by j)lanes to the two 

 points in which the quadric meets the opposite edge, the twelve 

 planes so obtained touch another quadric. But it may happen that 

 this new quadric degenerates into two points, say S and S' ; then, 

 with a proper choice of notation, the four lines AU, BV, CW are 

 concurrent in a point S, and the four lines A'U', B'V, CW con- 





