VII] OF THE BEE'S CELL 331 



It is required to find the inclination of the faces forming the 

 trihedral angle at V to the axis, such that the surface of the 

 figure may be a minimum. 



Let the angle NVX, which is half the solid angle of the prism, 

 = 6 ; the side of the hexagon, as AB, = a ; and the height, as 

 Aa, = h. 



Then, AC = 2a cos 30° = a^/3. 



And VX = a/sin 6 (from inspection of the triangle LXB) 



Therefore the area of the rhombus VAXC = a^\/3/2 sin 6. 



And the area of AabX = a/2 {2h - |FZ cos d) 

 = a/2 {2Ji-al2.cotd). 



Therefore the total area of the figure 



= hexagon abcdef + 3a [ 2h — ^ cot ^ ) + 3 „ . 



Therefore ^ (^""^^^ - ^""^ f ^-^ a/S^os^N 

 inereiore — ^^ - ^ l^sin^ ^ ' sin^ ^ / ' 



But this expression vanishes, that is to say, d (Area)/(^^ = 0, 

 when cos 6 - 1/^3, that is when d = 54° 44' 8" 



= 1 (109° 28' 16"). 



This then is the condition under which the total area of the 

 figure has its minimal value. 



That the beautiful regularity of the bee's architecture is due 

 to some automatic play of the physical forces, and that it were 

 fantastic to assume (with Pappus and Reaumur) that the bee 

 intentionally seeks for a method of economising wax, is certain, 

 but the precise manner of this automatic action is not so clear. 

 When the hive-bee builds a solitary cell, or a small cluster of cells, 

 as it does for those eggs which are to develop into queens, it makes 

 but a rude production. The queen-cells are lumps of coarse wax 

 hollowed out and roughly bitten into shape, bearing the marks of 

 the bee's jaws, like the marks of a blunt adze on a rough-hewn log. 

 Omitting the simplest of all cases, when (as among some humble- 

 bees) the old cocoons are used to hold honey, the cells built by 

 the "soHtary" wasps and bees are of various kinds. They may 

 be formed by partitioning off httle chambers in a hollow stem; 



