160 Professor Baker, On the reduction of homography 



This is then the condition for the existence of tetrahedra with 

 faces touching S of which two pairs of opposite edges he on Sq, 

 or equally, for the existence of tetrahedra with vertices on Sq, of 

 which two pairs of opposite edges lie on S. Or say, it is the condi- 

 tion for S to be tetrahedrally in- 

 scribed in Sq, or for Sq to be tetra- 

 hedrally circumscribed to S. 



Supposing the quadric S to be 

 in the specified relation to Sq, there 

 will, corresponding to each of the 

 three ways of pairing the roots of 

 the equation | /Sq — A;S | = 0, be two 

 figures such as that here indicated. 

 Here p, q are two transversals of 

 Sq, touching S, say at the points 

 X, y; 6 is an axis of Sq associated 

 with p, q ; and the Unes xz, xt and 

 yz, yt are the two generators of S 

 at the points x, y. The tetrahedron 

 of which the vertices are the points 

 {6, p), (9, q), each taken doubly, of 

 which the faces are the planes {d,p), 

 [6, q), each taken doubly, of which 

 the edges are the fines p, q and the 

 fine 6 taken four times over, has 

 its edges p, q, 9, 9 as generators of 

 Sq while its faces touch ;S; the tetrahedron x, y, z, t has its vertices 

 on Sq, and two pairs of opposite edges of it are generators of S. 

 Referred to {x, y, z, t) the quadric S may be taken to be 



xy — zt = 0, 



and the quadric Sq to be 



2hxy + x{gz+ ut) + y {fz + vt) = 0. 



Then the equation | aSq — A>S | = has the roots 



X„X,^h±{h^-[{fu)i + {gv)if}i, 



X^,X, = h±{h^-[{fu)^-{gv)if}i 



for which Aj + Ag = A3 -f A4, The axes of Sq are given by 



{h-9)x +fz + vt = 0, {h + e)y + gz + ut=0, 



and the transversals by 



X = py, hx + hpy + {/ + gp) z + {v + up) t = 0; 



the plane 



{h-9)x +fz + vt + p[{h + 9)y + gz + ut] = 0, 



