PARTITION OF A PARALLELEPIPED INTO TETRAHEDRA. 717 



The sexpartite division without new corners. Here we have thirteen genera, which 

 we shall consider in order — 



1st, 3L,3l\ The only variation which we can make in this case consists in taking 

 different body-diagonals as the axis. There are four body-diagonals, therefore there are 

 four ways in which this complex can be put into the box, and therefore four different 

 species in this genus. 



2nd, I,A,2L,2r. Here there is only one I. When its position is fixed, the position 

 of all the other tetrahedra is fixed. As the I can be placed in twenty-four positions, 

 there are twenty-four species in this genus. 



3rd, || 2l,2A,L,r. Here the position of the one I fixes that of the other I, for they 

 have the same body-diagonal, and their singular corners are at opposite ends of this 

 body-diagonal. When these two are fixed all the other tetrahedra are fixed, for the L 

 lies between the two I's on the one side, and the T between them on the other side, and 

 the position of each of the A's is, of course, fixed by that of the corresponding I. To 

 take an example, let AABC be one of the I's, the other is necessarily AABC, the L is 

 AABC and the V AABC, the two A's being DABC and DABC. 



The number of species in this genus is therefore the number of pairs of positions of I 

 of the form A ABC, A ABC. This number is obviously twelve, three for each body- 

 diagonal, — for the body-diagonal AA for instance, AABC,AABC ; AACD,AACD ; 

 AADB,AADB. 



4th, ff 2l,2A,L,r. Before considering this genus it will be convenient to take 



5th, 3 1, 3 A. Here the circumscribing sphere is divided into three lunes of 120°, each 

 of which corresponds to an IA. The singular corners of the three I's are all at the same 

 end of the axis, and therefore there are two different positions for each body-diagonal, 

 that is, eight positions in all, or eight species of this genus. 



Keturning to 4th, H 2l,2A,L,r, we see that in this form we have an IA of 3l,3A, 

 replaced by an LX pair. And in each position of 3l,3A, it may be any one of the IA's 

 that is so replaced. Each species, therefore, of 3 1, 3 A gives rise to three species 

 of H 2l,2A,Lr ; there are therefore twenty-four species of this genus. 



We now have to consider the biaxial forms ; and we have first to notice an essential 

 difference between the uniaxial and the biaxial forms. In the former every internal face 

 of one tetrahedron is covered by a single internal face of another tetrahedron. Thus, the 

 internal face of a A is covered by the equal internal face of an I, and the internal face of 

 an L, and similarly what we may call the " L" internal face of an I, are covered by 

 the equal and oppositely -placed internal face of an I\ or by what we may call the " T '' 

 internal face of an I. 



In the biaxial forms the case is quite different ; here we have always, in the plane 

 containing the two axes, either two L faces belonging to two tetrahedra on one side of 

 that plane covered by two L faces belonging to two tetrahedra on the other side of it, or 

 two r faces similarly covered by two T faces, as shown in fig. 30. The two L faces 

 (or the two T faces) may, of course, be either two internal faces of two L's (or two T's) or 



