7 IS PROFESSOR CRUM BROWN ON THE 



a face of an L (or V) and an " L " (or " r ") face of an I. It is therefore obvious that 

 when an L and a T unite to form an IA they must be on the same side of the plane 

 containing the axes. 



6th and 7th, 4L,2r, and 4r,2L. Each of these is completely determined in position 

 if the plane containing the axes is fixed. Thus with axes AA and BB we have in the 

 form 4L,2l\ on the one side of the plane of the axes, the tetrahedra, L = AABC, 

 r = AACD and L = AiVDB ; and on the other side of the plane the tetrahedra, L = BBAD, 

 r = BBDC, and L = BBCA. With the same axes we have in the form 4r,2L, on the one 

 side T = BBAC, L = BBCD, and T = BBDA ; and on the other side T = AABD, L = A ADC, 

 and T = AACB. 



As there are four body-diagonals, they may be taken two at a time as axes in six 

 different ways, and there are therefore six species in each of these genera. 



8th and 9th, IA,3L,r, and IA,3r,L. Each of these contains one I, and the I can 

 occupy any one of twenty-four positions in the cubical box, and the position of the I 

 fixes the positions of the other tetrahedra. There are therefore twenty-four species in 

 each of these genera. This can be shown in another way. Each species of 4L,2T (or of 

 4l\2L) can give rise to four species by the replacement of an LX pair by IA. For this 

 replacement may take place on either side of the plane of axes, and on either side in two 

 ways, the T (or L) uniting to form IA with the one or with the other of its L (or T) 

 neighbours. As there are six species of each of the genera 4L,2r and 4T,2L, there are 

 twenty-four species of each of the genera IA,3L,r and IA,3r,L. 



10th, 11th, 12th, and 13th. || 2(IA),2L, || 2(IA),2r, \\ 2(IA),2L, and || 2(IA),2r. 

 These forms are derived from 4L,2T and 4T,2L, by replacing an Lr pair on each side of 

 the plane of the axes by IA. 



In each form there are two ways on each side of the plane of the axis in which the 

 replacement of an Lr pair can take place, and either way on the one side can go with either 

 way on the other side. There are, therefore, for a given pair of axes, four positions for 

 2(IA),2L and four for 2(IA),2r. Of these half belong to the parallel and half to the non- 

 parallel forms. Thus for the axes AA and BB and the forms 2(1 A), 2L we have, on the 

 CI) side of the plane of the axes, the two arrangements: — I = BBAD, A = CBAD, 

 L = BBAD, and I = BBCA, A = DBCA, L = BBCA ; and on the other, the CD side of the 

 plane, the two arrangements :— I = AADB, A = CADB, L = AADB, and I = AABC, 

 A = DABC, L = AABC. 



Calling these arrangements, each of which makes up half of the cube, C, D, C and D, 

 after the singular corner of the A in each, we see that C and C or D and D go together 

 to form ||2(IA),2L, and C and D, or C and D go together to form |j 2(1 A),2L. In a 

 similar way for 2(IA),2r we have the four arrangements : — On the CD side, I = AABD, 

 A = CABD, T = AABD, and I = AACB, A = DACB, r = AACB, and on the CD side 

 I - BBDA, A = CBDA, r = BBDA, and I = BBAC, A = DBAC,r = BBAC. Of these C 

 and C, or I) and D go together to form || 2(IA),2r, and C and D or C and D go together 

 to form H 2(IA).2l\ Thus, for each pair of body-diagonals as axes there are two positions 



