156 Mr Hanumanta and Professor Baker, On the generation of 



point of the quadric into another point lying on the same q- 

 generator; such a transformation will interchange the 2>generators 

 among themselves, save for two p-generators, each of which will 

 be unchanged. We may call such a transformation a p-transforma- 

 tion. Let ABCD be the tetrahedron arising from XYZT by any 

 such transformation; let PQRS be the tetrahedron arising from 

 ABCD by inversion from the point T. Thus PQRS may also be 

 described as arising from XYZT by the g-- transformation which 

 effects the same transformation, of the points of the section of the 

 quadric by the plane XYZ, as does the transformation by which 

 ABCD is derived from XYZT; in other words if the parameters 

 for the f, g'-generators be chosen so as to be the same for the two 

 generators which intersect at a point of the section by the plane 

 XYZ, then PQRS is derived from XYZT by the 5-- transformation 

 expressed by the same equation, connecting the parameters of the 

 generators, as is the ^-transformation by which ABCD is derived 

 from XYZT. 



Then take the inverses of PQRS respectively from X, Y, Z, T, 

 and let them be D^C-^B^A^, C^D^A^B^, B^A^D^C^, ABCD. The 

 tetrahedra A^BjC\Dj^, A2B2C2D2, A^B^C^D^, ABCD are then such 

 that every two are mutually inscribed. And conversely given any 

 such four tetrahedra, the transformation by which XYZT, PQRS 

 are determined is definite. The tetrahedron XYZT is unique, being 

 defined by the fact that the edges YZ, XT are the diagonals of 

 the skew quadrilateral formed by (1) the two g'-generators meeting 

 the edges BC, AD, (2) the two ^J-generators which are the common 

 transversals of AA^^, BB^, CC-^, DD^, while the other two pairs of 

 edges are similarly obtainable. The tetrahedron PQRS has however 

 been obtained from XYZT and ABCD in an unsymmetrical way, 

 and there are three other possibilities. Let P-^Q^R^S^ be the tetra- 

 hedron obtained from ABCD by inversion from X. Then D^C^B^A^, 

 C^D^A^B^, B^AJD^C^, ABCD are obtainable from P-^Q-Jl-^S-^ by 

 inversion respectively from TZYX. So if P2^2^2*^2 be obtained 

 from ABCD by inversion from Y, the same four tetrads of points 

 are obtained by inversion of P^Q^^-S^ respectively from ZTXY ; 

 and if P^Q^R^S^ be obtained from ABCD by inversion from Z, the 

 same four tetrads are obtained by inversion of PgQ^R^S^ respec- 

 tively from YXTZ. But Pj^Q^R^S^, P^Q^RzSz, P3Q3R3SS are ob- 

 tainable from PQRS by harmonic inversion in two lines in each 

 case, respectively YZ, TX; ZX, TY and XY, TZ. 



Taking then the first case, where PQRS is used, since inversions 

 in the vertices T, X, in succession, in either order, are together 

 equivalent to harmonic inversion in the opposite edges YZ, TX, 

 we may also say that the tetrads D^C^B^A^, C^D^A^^.^ B^A^D^C^ 

 are obtainable from ABCD by harmonic inversion resfectively in the 

 fairs of edges YZ, TX; ZX, TY; XY, TZ. And we have remarked 



