714 PROFESSOR CRUM BROWN ON THE 



singular corners of the I and of the A) cuts the I A into an L and a F, so that we may as 

 well call this figure an Lr as an IA. We thus see that in any complex, IA may be 

 replaced by a pair consisting of an L and a F (see figs. 7, 8, 9). In an LF pair there are 

 obviously three corners of the L which coincide each with a corner of the F. Two of 

 these are the corners at the ends of the common body-diagonal, the other occupies tin; 

 place of the singular corner of the A in the IA by which the Lr can be replaced. We 

 may call this the A corner of the Lr or IA. In considering the sexpartite divisions of 

 the cube we may therefore begin with those containing only L's and T's, and then derive 

 the rest from these by putting IA in place of Lr. 



These complexes containing only L's and T's may be called central forms, as everv 

 tetrahedron in them has an edge bisected at the centre of the cube. The simplest central 

 form is that in which all the tetrahedra, three L's and three T's, meet in one body- 

 diagonal. It is obtained by cutting the cube by three planes, each passing through the 

 same body-diagonal and two parallel edges. It is shown in fig. 10. It will be seen that 

 any one of the three planes mentioned above divides the cube into two halves. These 

 halves are not similarly divided ; one is divided into two L's with a T between them, and 

 the other into two T's with an L between them. But as each of them is exactly a half 

 cube, and as they are externally precisely alike, a whole cube can be made up quite as 

 well of two of the first kind or of two of the second kind as of one of each. We thus 

 obtain two other central forms consisting of 4 L's and 2 T's, and of 4 T's and 2 L's 

 respectively (figs. 11, 12). These three central forms will perhaps be more easily imagined, 

 in the absence of the models which were exhibited to the Society, by supposing the 

 cube surrounded by a circumscribed sphere, the surface of which is divided into lunes of 

 00°, each of which corresponds to an L or to a F. The axis in reference to which the 

 meridians are drawn is a body-diagonal of the cube, and each meridian passes through 

 one of the non-polar corners of the cube. A lune of 60° corresponds to an L if it has a 

 corner of the cube in the northern part of its western and one in the southern part of its 

 eastern meridian, and to a F if it has a corner of the cube in the southern part of its 

 western and one in the northern part of its eastern meridian. A lune of 120° corresponds 

 to the figure I A or LF. 



In the central form, 3L,3F, all the meridians are drawn in reference to one body- 

 diagonal ; in the central forms, 4L,2T and 4F,2L, there are two axes, both, of course, 

 body-diagonals of the cube, and the plane containing the two axes is the plane cutting 

 the cube into the two halves referred to above. Stereographic projections of one half of 

 the sphere in the three cases 3L,3F, 4L,2r, and 4r,2L, are shown in figs. 13, 14, 15. 

 The point from which the projection is taken is on the sphere half-way between A and 

 li. The forms are shown in figs. 10, 11, 12. It will, of course, be seen that the two 

 biaxial forms (as we may call 4L,2F and 4T,2L, in distinction from the uniaxial 3L,3T) 

 arc cnantiomorph, and accordingly all their derivatives occur in enantiomorph pairs. 



We shall consider in the first place the uniaxial forms, that is to say, the derivatives 

 of3L,3l\ 



