446 



NA TURE 



[March 8, 1894 



and three quadrilateral spaces in the manner more particularly 

 explained in § 6 below. It is clear, or at all events I shall 

 endeavour to make it clear by fuller explanations and illustrations 

 below, that the figure thus constituted fulfils our definition (§ i) 

 of the most general form of cell fitted to the particular homo- 

 geneous assemblage of points corresponding to the parallelepiped 

 with which we have commenced. This will be more easily 

 understood in general, if we first consider the particular case of 

 parallelepipedal Y>3iXi\\.iQmv\g, and of the deviations which, without 

 altering its corners, we may arbitrarily make from a plane-faced 

 parallelepiped, or which we may be compelled by the particular 

 ff^ure of the molecule to make. 



§ ■;. Consider, for example, one of the trees of § 2, or if you 

 pleaje a solid of less complex shape, which for brevity we shall 



(Fig. 7, of §9.) 



call s, being one of a homogeneous assemblage. Let P be a 



point in unoccupied space (air, we shall call it for brevity), 



which, for simplicity we may suppose to be somewhere in the 



immediate neighbourhood of s, although it might really be any- 

 where far off among distant solids of the 



assemblage. Let pa, pb, pc be lines parallel 



to any three Bravais rows not in one plane, 



and let A, B, c be the nearest points cor- 

 responding to P in these lines. Complete a 



parallelepiped on the lines pa, pb, pc, and let 



QD, QE, OF be the edges parallel to them 



through the opposite corner n. Because 



of the homogeneousness of the assemblage, 



and because a, b, c, D, e, f, n are points 



corresponding to p which is in air, each of 



those seven points is also in air. Draw any 



line through air from P to A and draw the 



lines of corresponding points from B to F, d 



to Q, and c to E. Do the same relatively 



to PB, AF, EQ, CD ; and again the same 



relatively to PC, ae, fq, bd. These twelve 



lines are all in air, and they are the edges of 



our curved-faced parallelepiped. To describe 



its faces take points infinitely near to one 



another along the line PC (straight or curved 



as may be) : and take the corresponding 



points in bd. Join these pairs of correspond- 

 ing points by lines in air infinitely near to one 



another in succession. These lines give us 



the face pbdc. Corresponding points in ae, 

 FQ, and corresponding lines between them 

 give us the parallel face afqe. Similarly we 

 find the other two pairs of the parallel faces of 

 the parallelepiped. If the solids touch one 

 another anywhere, either at points or through- 

 out finite areas, we are to reckon the interface 

 between them as air in respect to our present 

 rules. 



§ 6. We have thus found the most general 

 possible parallelepipedal partitioning for any 

 given homogeneous assemblage of solids. 

 Precisely similar rules give the corresponding 

 result for any possible partitioning if we first 

 choose the twenty-four corners of the tetra- 

 kaidekahedron by finding six tetrahedrons and 

 giving them arbitrary translatory motions ac- 

 cording to the rule of § 4. To make this clear it is only now 

 necessary to remark that the four corners of each tetrahedron 

 are essentially corresponding points, and that if one of them is 



NO. I 27 I, VOL. 49] 



in air all of them are in air, whatever translatory motion we give 

 to the tetrahedron. 



§ 7. The transition from the parallelepiped to the tetrakaide- 

 kahedron described in § 4 will be now readily understood if we 

 pause to consider the vastly simpler two-dimensional case of 

 transition from a parallelogram to a hexagon. This is illustrated 

 in Figs. I and 2 ; with heavy lines in each case for the sides of 

 the hexagon, and light lines for the six of its diagonals which 

 are sides of constructional triangles. The four diagrams show 

 different relative positions in one plane of two equal homochirally 

 similar triangles abc, a'b'c' ; oppositely oriented (that is to say, 

 with corresponding lines ab, a'b' parallel but in inverted direc- 

 tions). 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 cor- 

 responding 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 demon- 

 stration are superfluous. 



§ 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, mn, uv. The piece M 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 



Fig. 



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 



