Professor Baker, On the Hart circle of a spherical triangle 117 



current in another point S'. That this should be so is a necessary 

 and sufficient condition that the four sections of the quadric by 

 the faces of the tetrahedron should have four common tangent 

 planes. The condition may be stated in another form; take on 

 the edge OX, the point Ai separated harmonically from A by 

 and X, and the point Ai separated harmonically from A' by 

 and X; in the same way take on each edge of the tetrahedron 

 the harmonic conjugates, with regard to the vertices of the tetra- 

 hedron lying on that edge, respectively of the intersections of the 

 quadric with that edge. The twelve new points so obtained lie on 

 another quadric, which we may describe as the harmonic conjugate 

 of the original in regard to the tetrahedron. The condition in 

 question then is that the harmonically conjugate quadric should 

 break up into two planes, say a and a'; these will be the polar 

 planes of >S and S' in regard to the original quadric. 



We may illustrate this condition by applying it to the (Feuer- 

 bach) case of three sections of the quadric which are concurrent 

 on the quadric, say in 0. The fourth section of the quadric touched 

 by the four common tangent planes of the three given sections 

 OYZ, OZX, OXY is then constructed as follows: on the plane YOZ 

 take the line p through 0, harmonically conjugate with respect to 

 OY, OZ, to the line in which the plane YOZ is met by the tangent 

 plane of the quadric at 0; let this line p meet the quadric again 

 in P; obtain the points Q, R of the sections ZOX, XOY in a similar 

 way. The plane PQR is the fourth plane required. In this case 

 one of the planes a, a' is the tangent plane at 0. 



§ 2. We may obtain a direct verification of the sufficiency of 

 the condition in general by using it to obtain any one of the eight 

 (Hart) sections m which can be associated with three given sections 

 YOZ, ZOX, XOY, so as to form four sections with four common 

 tangent planes. Let the quadric, referred to YOZ, ZOX, XOY and 

 the polar plane of 0, have the equation 



ax'^ + by^ + cz^ + ^fyz + 2gzx + 2hxy = t^ ; 



with an arbitrary choice of the signs of Va, Vb, Vc, take 



u = h (/ -f Vb Vc), V = -| {g + Vc Va), w = | {h + Va Vb), 



and then I, m, n so that 



mn = u, nl = v, hn = w; 



the eight planes required are then expressed by 



Ix + my + nz — t^ = 0. 



It is at once seen that this follows from the condition stated above. 

 If we introduce A, fi, v so that 



/ = Vb Vc cos X, g ^ Vc Va cos /x, h = Va Vb cos v. 



