1894.] 



On Homogeneous Division of Space. 

 FIG. 6. 



lengths of face- diagonals having ends in P or Q (three pairs of equal 

 parallels) ; and the length of the chosen body-diagonal PQ. (Any 

 one of these seven is, of course, determinable from the other six if 

 given.) 



In the diagram, fig. 7, full lines show the edges of the primitive 

 parallelepiped, and dotted lines show the body-diagonal PQ and two 

 pairs of the face- diagonals, the other pair of face-diagonals (PF, QC), 

 not being marked on the diagram to avoid confusion. Thus, the dia- 

 gram shows, in the parallelograms QDPA and QEPB, two of the three 

 cutting planes by which it is divided into six tetrahedrons, and it 

 so shows also two of the six tetrahedrons, QPDB and QPEA. The 

 lengths QP, QD, QB, QF are found in the edges of every one of the 

 six tetrahedrons, the two other edges of each being of two of the three 

 'engths QA, QB, QC. The six tetrahedrons may be taken in order of 



