71(5 PROFESSOR CRUM BROWN ON THE 



II. When the Parallelepiped is General. 



By a general parallelepiped I mean one in which, we do not assume any particular 

 value of any plane angle, and therefore no right angle, and no equality of any parts of 

 the parallelepiped, except such as are necessarily equal by the definition of the parallele- 

 piped. In a general parallelepiped the eight corners are, therefore, all different from one 

 another. It is only in particular cases that we have interchangeable corners. No doubt, 

 the opposite corners (such as A and A) are equal, but they are not interchangeable. The 

 opposite corners are enantiomorph, mirror-images of each other. A will no more fit into 

 a mould of A than a right hand will fit into the mould of an otherwise perfectly equal 

 left hand. Therefore, all the distinct positions of the tetrahedral partitions of the cube 

 correspond to essentially different partitions of the general parallelepiped. A cube can 

 be converted into any parallelepiped by appropriate homogeneous strains, and, conversely, 

 any parallelepiped into a cube. Such strains do not change the volume-ratios of the 

 tetrahedra into which the figure has been divided, and parallel lines remain parallel, and 

 intersecting lines still intersect after as before the change of form. 



Each of the fourteen partitions of the cube may therefore be taken as the type of a 

 genus of partitions of the parallelepiped ; and we may use the models and diagrams which 

 were made for the case of the cube for the case of the general parallelepiped, if we keep in 

 mind that what was with the cube only a difference of position is in the case of the parallel- 

 epiped an essential difference of form. Indeed, it is not necessary that we should suppose 

 even a difference of form, for we may suppose any other difference which would make the 

 eight corners non-interchangeable. For instance, we might suppose our cube to be com- 

 posed of a heterogeneous material, say of gold and silver, so that the ratio of the two 

 metals varied with the distance from a point not equidistant from any two corners. Here 

 two A's would indeed be of exactly the same form, but they would be neither chemically 

 nor commercially interchangeable. 



What we have now to do, therefore, in order to ascertain the number of different 

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

 new corners, is to count the number of ways in which the fourteen divided cubes can be 

 put into a cubical box the corners of which are distinctively noted. 



Taking first the quinquepartite division 0,4 A, we see that & can occupy two 

 positions in the cubical box, viz., ABCD and ABCD. Each of these absolutely 

 determines the positions of the four A's, so that there are two, and only two, quinque- 

 partite divisions of the general parallelepiped. Neglecting the condition forbidding new 

 corners, each of these quinquepartite divisions gives rise to six sexpartite divisions with 

 six tetrahedra of equal volume. For the can in each case be cut into two equal- 

 volume tetrahedra by a plane passing through an edge and bisecting the opposite edge, 

 and, in the parallelepiped, Q has six different edges. 



