230 On the Mat hematics of Bees' Cells. 



Omitting a constant term and a constant factor, the quantity 

 to be made a maximum is 



provided that the rhombuses have the same thickness as the 

 prism-walls. If they are ?z times as thick as the walls, the 

 quantity to be made a maximum is 



^(=zh--n^^/{s'-\-U') ; 

 whence, putting 77"^? we find s^ = {I27i''^ — 4) h^\ 



or 



giving 



h'^ s' + h' s'-'2h 



127i2-4~l ~ 12n2-a'~ J2?^2-6' 



The usual calculation assumes n = l, giving 

 cos 0=1, cos<^=^. 



The trihedral angles, of which there is one at each apex> 

 and one at the end of each of the shorter prism-edges, are 

 each composed o£ "3"plane angles whose cosine is — A, the 

 inclinations of their planes being 120°. The form thus 

 deduced is regarded as the normal form of bees' cells. 



It is closely related to the most compact system of piling 

 of equal spheres. In this system each sphere touches 3 

 spheres in the layer below, 3 in the layer above, and 6 in 

 its own layer. If we omit the 3 upper or the 3 lower, the 

 tangent planes at the 9 remaining points of contact represent 

 the 9 walls which bound a cell. 



Again, the 9 walls (including 3 pairs of parallel walls) 

 are perpendicular to the 6 edges of a regular tetrahedron; 

 and the 12 lines of junction of these walls (including 3 

 pairs of parallels and 1 set of 6 parallels) are perpendicular 

 to the 4 faces of the same tetrahedron. 



According to observations cited by Darwin, the actual 

 value of n averages about |. This gives 



cos = -j^g , cos (f) = x/^^j 



for minimum consumption of material. 



