PARTITION OF A. PARALLELEPIPED INTO TETRAHEDRA. 715 



By the replacement of one LT pair by an IA, we have the form I,A,2L,2r. There 

 is only one form with these tetrahedra, shown in fig. 16, because all the Lr pairs are 

 similarly situated in the central complex 3L,3r. But there are two ways in which two 

 Lr pairs can be replaced by two IA's. For the two pairs replaced, and, of course, the two 

 IA's replacing them, may be either contiguous, or separated from one another by an L on 

 the one side and a T on the other. "We thus have two different forms, 2(IA),LI\ 

 They are shown in figs. 17, 18, and may be distinguished as j| 2(IA),Lr and ||2(IA),Lr, 

 as in the first, where the IA's are contiguous, the plane of the equilateral triangle of the 

 one IA is inclined to that of the other, while in the second form, where the two IA's are 

 not contiguous, the plane of the one equilateral triangle is parallel to that of the other. 

 There is only one form, 3(IA), produced by replacing all three Lr pairs by IA's. It is 

 shown in fig. 19. 



Turning now to the biaxial forms, we see that it is sufficient to describe one half of 

 them, because, on account of the enantiomorphism, everything that is true of 4L,2r and 

 its derivatives can be made to apply to 4r,2L and its derivatives by changing L into T 

 and r into L. 



We shall therefore consider the derivatives of 4L,2F only. 



By replacing one Lr pair by IA, we obtain the form IA,3L,I\ Of this there is only 

 one, as any Lr in 4L,2T can be brought, by turning the form about into the position of 

 any other. But, as in the uniaxial series, so here, there are two essentially different 

 ways in which tiro LPs can be replaced by two IA's. As to the two Ps of the LPs, 

 there can be no dubiety, for there are only two in the form, but as each T lies between 

 two L's, we may have the two Ps paired with L's, so that the two A corners have the 

 same letter and opposite signs, or the same sign and different letters ; and thus we have 

 two forms, || 2(IA),2L and {{ 2(IA),2L, as with the uniaxial forms with 2(IA). These 

 biaxial forms are all shown in figs. 20-27. 



We have in all, then, the following divisions of the cube without new corners : — 



Quinquepartite. Q, 4A. One form. 



Sexpartite. Uniaxial. 3L,3P (1A),2L,2P j{2(IA),LP || 2(IA),LP 



3(IA). Five forms. 

 Biaxial. 4L,2P (LA),3L,P {{2(IA),2L. II 2(IA),2L. 

 4l\2L. (IA),3l\L. ff2(IA),2l\ il 2(IA),2P 

 Eight forms in four enantiomorph pairs. 



