712 Proceedings of Royal Society of Edinburgh. [sess. 
Closest Packing of one Homogeneous* Assemblage of Equal 
and Similar Globes or Ellipsoids. 
§46. Take our tetrahedron OPQR, and by homogeneous distor- 
tional strain convert it into an equilateral tetrahedron ABCD, of 
equal volume. Take four globes, of diameters equal to the edges of 
this tetrahedron and place them with their centres at its corner 
points A, B, C, D. Alter this assemblage of globes by homogeneous 
strain till their centres, ABCD, become again the corner points of the 
original tetrahedron, OPQR. The globes have now become ellip- 
soids. Dealing thus with the whole original homogeneous assemblage 
of points, we find a closest packed homogeneous distribution of 
equal and similar ellipsoids through space. 
§ 47. To find every possible closest packed homogeneous assem- 
blage of given equal and similar ellipsoids, take a tetrahedron of four 
equal globes. Choose any three mutually perpendicular directions, 
and, by elongations and shrinkages of the group parallel to these 
directions, convert each globe into an ellipsoid equal and similar to 
* There is another closest packing of globes or ellipsoids which has the same 
density as, and might without careful attention be mistaken for, the closest 
homogeneous packing. For simplicity think only of globes, and take a plane 
covered with globes touching one another in equilateral triangular order. 
Look at the accompanying diagram, fig. 6 of § (55) below, and see that there are 
two ways of placing a second layer on the first to continue the formation of an 
assemblage. The globes of the second layer may be placed, all of them over 
the black dots ( • ) or all of them over the white dots ( ° ). But having once 
chosen the position of the second layer there is no more freedom to choose in 
adding on layer after layer if we are to make a single homogeneous assemblage. 
Of the two positions which might be chosen for the third layer we must choose 
the one in which the globes are not over the globes of the first layer. The 
position of the fourth layer must be the one of which the globes are not over 
the globes of the second layer, but are over those of the first layer, and so on. 
If on the contrary we place the globes of the third layer over the globes of 
the first, the globes of the fourth layer over those of the second, and so on, we 
have a peculiar and symmetrical grouping which was first, so far as I know, 
described by Mr William Barlow ( Nature , December 20 and 27, 1883). This 
grouping is not one homogeneous assemblage. It consists of two homogeneous 
assemblages, one of them constituted by the first, third, fifth, seventh, &c. , 
layers ; the other the second, fourth, sixth, eighth, &c., layers. The considera- 
tion of this peculiar mode of grouping may be of great interest in the dynamical 
investigations to form the subject of my next communication to the R.S.E. 
(July 15), and, as Barlow has pointed out, may be of great importance in the 
theory of natural crystalline structure. I must, however, leave it for the 
present. 
