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



a plane of this latter form is 



/- cos la cos |v /r 



Va -.s + y V6 



cos |A 



- cos l-v cos ^A 



A cos |-jLt 



/-cos iA cos ia ^ ^ ,TT\. 



+ Z-VC — — -2r_ t = 0, (H) ; 



cos |v 



on the other hand a common tangent plane of the three given 

 sections in x = 0, ^ = 0, 2 = is at once found to be 



X Va cos {s- X) + yVh cos (s — /x) + 2 Vc cos {s — v) — t-^ = 0, (I) 



where s = ^ (A + yu, + v) ; and it is easy to see that the section (I) 

 touches the section (H) at the point of the plane (H) which lies on 



X Va : y^/h : zVc = p (q— rf : q (r — p)^ : r (p — qf, 



where, for brevity, p, q, r stand respectively for 



sin {s — A), sin (s — fj,), sin {s — v). I 



The four planes (I) which touch the section (H), as well as the 

 original sections inx = 0,y = 0,z = 0, are obtained from the above 

 equation by replacing A, yu,, v by ± A, ± /x, ± v, respectively. 



The eight sections (H) are obtainable from that above by re- 

 placing Va, Vb, Vc, A, /x, V respectively by 1 



{Va, Vb, Vc, A, fji, v), (— Va, Vb, Vc, A, 77 + ix,7t -\-v), 



{Va, — Vb, Vc, X + TT, ix,v + tt), {Va, Vb, — Vc, X + tt, jx + tt,v) 

 together with those obtainable from these by changing the sign 

 of ^1. 



§ 3. The following result gives a construction for the position, 

 upon the section, i, of the quadric by the plane (I), of the point 

 in which this section is touched by the plane (H). Upon i we have 

 three points, its contacts with the sections in a? = 0, ?/ = 0, s = 0; 

 we also have two points, namely those in which i is met by the 

 plane from to the intersection of the planes ABC, A'B'C , which 

 plane is at once found to have the equation 



x Va + y Vb + s Vc = 0. 

 The point to be constructed is the apolar complement of the two 

 latter points in regard to the three former points. This result may 

 be made clearer perhaps by stating it for a sphere in Euclidian 

 geometry: If D, E, F be the mid-points respectively of the sides 

 BC, CA, AB of a spherical triangle, the planes of the great circle 

 arcs EF, BC give a diameter, and the three diameters so obtained 

 are coplanar; let I, J denote the intersections of their plane with 

 the inscribed circle of the triangle ABC; let P, Q, R be the points 

 of contact of this inscribed circle with the sides BC, CA, AB. Then, 



