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XXXI. — On the Partition of a Parallelepiped into Tetrahedra, the Corners of which 

 Coincide with Comers of the Parallelepiped* (With Two Plates.) By Professor 

 Crum Brown. 



(Read 5th and 19th March 1894.) 



I. When the Parallelepiped is a Cube. 



It will be convenient first to fix a form of notation for the corners of the cube. With 

 the origin within the cube and rectangular co-ordinates parallel to three edges meeting in 

 a corner, the eight corners lie each in one of the eight octants, and may conveniently and 

 symmetrically be represented thus : — 



+ + + 



A, 







A 



+ — 



B , 



- + + 



B 



- + - 



C, 



+ - + 



C 



— + 



D, 



+ + - 



D. 



Or, calling any corner A, the corners distant »J 2 from A (the length of the edge 

 being l) and taken positively, i.e., contrary to the watch-hand way, as seen from A, are 

 B, C, I) (see figures). 



So that, passing from corner to corner along an edge, we change both letter and sign, 

 along a face diagonal we change letter but not sign, along a body diagonal we change 

 sign but not letter. 



The forms of tetrahedra which can be cut out of a cube without making new corners 

 are the following five, each of which is noted by one of its positions in the cube, — 



1. ABCD, the regular tetrahedron, the edges coinciding with the non-parallel face- 

 diagonals of the cube. 



2. ABCD, a tetrahedron with three contiguous half -faces of the cube and for its fourth 

 face the equilateral triangle whose side is the face-diagonal. Its opposite edges are an 

 edge and a face-diagonal of the cube. We may call the corner which differs in sign 

 from the others, the singular corner. It is an undivided corner of the cube. 



These are all the forms with all .four letters; for four corners with all four letters, 

 two of one and two of the other sign, such as ABCD, lie all in one plane and 

 represent the four corners of a face of the cube. 



3. AABC, a tetrahedron two faces of which are the scalene triangles with the edge, 

 the face-diagonal and the body-diagonal of the cube for sides, one face a half-face of the 



* This paper arose out of a conversation with Lord Kelvin last December. He was interested in one set of tetra- 

 hedral partitions of the general parallelepiped, of which he has since made use in his discussion of the homogeneous 

 partition of space, and thus interested me in the general question which he suggested as worthy of detailed 

 investigation. 



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



