PARTITION OF A PARALLELEPIPED INTO TETRAHEDRA. 719 



of each of the parallel forms, and two of each of the non-parallel forms, and therefore 

 twelve species of each of the four genera. 



Of the thirteen genera of sexpartite divisions without new corners we have therefore 

 the following number of species : — 1st, 3L,3F, four species; 2nd, (IA),2L,2r, twenty-four 

 species ; 3rd, || 2(IA),L,r, twelve species ; 4th, ff 2(IA),L,r, twenty-four species ; 5th, 

 3(IA), eight species — that is seventy-two species of the five uniaxial genera ; 6th, 4L,2r, 

 six species; 7th, 4r,2L, six species; 8th, (IA),3L,r, twenty-four species ; 9th, (IA),3F,L, 

 twenty-four species ; 10th, || 2(IA),2L, twelve species ; 11th, || 2(1 A), 2r, twelve species ; 

 12th. ff- 2(IA),2L, twelve species ; 13th, {{ 2(IA), 2r, twelve species — that is 108 species ; 

 of the eight biaxial genera. In all 180 species of the thirteen genera of sexpartite 

 divisions without new corners. If we add the two quinquepartite species we have 182 

 ways in which a general parallelepiped can be cut into tetrahedra without making new 

 corners, or if we add the twelve species of equal-volume sexpartite divisions with new 

 corners derived from the quinquepartite species, we have 192 ways in which a general 

 parallelepiped can be cut into six tetrahedra of equal volume. 



In counting the number of ways in which a parallelepiped can be cut into tetrahedra, 

 the only properties of the parallelepiped which have been used are that it is bounded by 

 six quadrilateral faces, and that its four body-diagonals intersect in a point. The special 

 character of a parallelepiped, that it has three sets of edges, the four edges in each set 

 being parallel and equal in length, has been used only to show that the six tetrahedra 

 into which the parallelepiped is divided are equal in volume. Everything, therefore, that 

 has been said of the divisions of the general parallelepiped into tetrahedra is true of a 

 hexahedron, the four body-diagonals of which intersect in a point, except the equality in 

 volume of the tetrahedra, and the parallelism of the two IA planes in such forms as 

 2(IA),L,r. Professor Chrystal has pointed out to me that in such hexahedra, instead of 

 three sets of parallel edges as in the parallelepiped, there are three sets of concurrent 

 edges, the four edges in a set meeting, when produced, in a point, and that an ordinary 

 perspective projection of a parallelepiped is an orthogonal projection of such a hexa- 

 hedron. 



VOL. XXXVII. PART IV. (NO. 31) 5 Q 



