378 



Professor G. D. Liveing 



[May 15, 



in the figure, and may place six others, touching it, and resting in the 

 six adjacent triangular spaces which are black in the figure. Above 

 these we can again place three more so as to touch the first. If we 

 complete the pile we get a triangular pyramid, as in Fig. 2. Or we 



Fig. 2. 



may begin by arranging balls on the flat, as in Fig. 3, so that each is 

 touched by four others. We may then place one ball so as to rest on 

 four, such as a, Z), c, d, in the figure. Then place four others, touching 

 it, in the four adjacent square-shaped openings which are shaded in 

 the figure. Above these, in places corresponding to a, h, c, d, four 

 more may be placed so as to touch the first. If the pile be completed 

 it will form a four-sided pyramid, as in Fig. 4. 



Fig. 3. 



Fig. 4. 



Although this arrangement seems at first sight difierent from that 

 in Fig. 2, it is not so ; for it will be seen that in the faces of the 

 pyramid of Fig. 2 the arrangement is that of Fig. 3, while in the faces 

 of the pyramid in Fig. 4 the arrangement is that of Fig. 1. Fig. 2 

 is really part of Fig. 4 turned over on its side. 



Before proceeding to the packing of ellipsoids, let us consider 



