Mr Hanumanta and Prof. Baker, Generation of tetrahedra 155 



On the generation of sets of four tetrahedra of which any two 

 are mutualhj inscribed. By C. V. Hanumanta and Professor 

 H F Baker 



[Read 8 March 1920.] 



If from a point P the transversal be drawn to two given skew 

 lines, the point P' of this transversal harmonically separated from P 

 by these lines may be said to be obtained from P by harmonic 

 inversion in regard to them. If and cj be a given point and a 

 given plane, and on the line joining to an arbitrary point P 

 there be taken the point P' harmonically separated from P by 

 the point and the plane w, we may speak of P' as obtained from P 

 by harmonic inversion in regard to and w; in particular when w 

 is the polar plane of in regard to a given qua dric it may be 

 sufficient to speak of P' as the inverse of P in regard to 0, this 

 use of the term including the ordinary use when inversion in regard 

 to a circle is spoken of. 



When two tetrahedra ABCD, A^B^CJ)^ are such that the 

 points A, B, C, D, A-^, B^, Cj, Dj lie respectively on the planes 

 B^C^D^, C^A^D^, A^B^D^, A^B^C^, BCD, CAD, ABD, ABC, they 

 will be said to be mutually inscribed. When this is so the two 

 transversals of the four lines AA^, BB^, CC-^, DD^ are generators 

 of a certain quadric in regard to which both the tetrahedra are 

 self -polar (the two transversals of the four lines BC, AD, B^C^, A^Dj 

 being generators of this quadric of the other system). If another 

 tetrahedron 4 2-^2^2-^2' self -polar in regard to the same quadric, 

 be in- and circumscribed to ABCD, and such that the two trans- 

 versals of A A 2, BB2, CC2, DD2 belong to the same system of 

 generators of this quadric as do the transversals of AA^, ..., DD^, 

 then ABCD, A^B^C^D^ may be said to be mutually inscribed in 

 the same sense as are ABCD, A^B^CJ)^. It is well known that 

 there exist systems of four tetrahedra of which every two are 

 mutually inscribed in the same sense. 



The object of the present note is to point out that any such 

 four tetrahedra may he regarded as all derived from a single other 

 tetrahedron, by inversion of this in the vertices of a certain further 

 tetrahedron, taken in turn, each of these vertices being associated 

 with the opposite face of this tetrahedron in this process of inversion. 



More precisely we may state this result thus. Let a tetrahedron 

 XYZT be self -polar in regard to a certain quadric. Denote the 

 two systems of generators of this quadric as the p-system and the 

 g'-system. There exist linear transformations of space changing any 



