Lord Kelvin. [Jan. 18, 



FIG. 2. 



oriented (that is to say, with corresponding lines AB, A'B' parallel 

 but in inverted directions). The hexagon AC'BA'CB', obtained by 

 joining A with B' and C', B with C' and A', and C with A' and B', is 

 clearly in each case a proper cell-figure for dividing plane space 

 homogeneously according to the Bravais distribution of points defined 

 by either triangle, or by putting the triangles together in any one of 

 the three proper ways to make a parallelogram of them. The corre- 

 sponding operation for three-dimensional space is described in 4 : 

 and the proof which is obvious in two-dimensional space is clearly 

 valid for space of three dimensions, and therefore the many words 

 which would be required to give it formal demonstration are super- 

 fluous. 



8. The principle according to which we take arbitrary curved 

 surfaces with arbitrary curved edges of intersection, for seven of the 

 faces of our partitional tetrakaidekahedron, and the other seven 

 correspondingly parallel to them, is illustrated in figs. 3, 4, 5, and 6, 

 where the corresponding thing is done for a partitional hexagon 

 suited to the homogeneous division of a plane. In these diagrams 

 the hexagon is for simplicity taken equilateral and equiangular. In 

 drawing fig. 3, three pieces of paper were cut, to the shapes kl, win, 

 uv. The piece kl was first placed in the position shown relatively to 

 AC', and a portion of the area of one cell to be given to a neighbour 

 across the frontier C'A on one side was marked off. It was then 

 placed in the position shown relatively to A'C and the equivalent 

 portion to be taken from a neighbour on the other side was marked. 

 Corresponding give-and-take delimitations were marked on the fron- 

 tiers C'B and B'C, according to the form win; and on the frontiers BA', 

 AB', according to the form uv. Fig. 4 was drawn on the same plan 



