536 THE FORMS OF TISSUES [ch. 



A soap-bubble, or soap-film, assumes a minimal configuratiou 



instantaneously*, however small the saving of surface-area may be. 



But after learning that the bee's cell has undoubted minimal 



properties, we should hke to know what saving is actually obtained 



by substituting a rhomboidal pyramid for a plane base in the 



hexagonal prism. It turns out, after all, to be a small matter! 



The calculation was first made by Maclaurin and by Lhuiller, in 



both cases briefly but correctly. Lhuiller stated that the whole 



amount used in the bee's cell was to that required for a flat-topped 



prismatic cell of equal volume as 25 + V6 (or 2745) to 28, the 



saving being thus a little more than 2 per cent, of the whole quantity 



of wax required t- Glaisher recalculated the values, taking the cell 



part by part. Assuming, with Lhuiller, that the radius of the 



inscribed circle of the hexagon is to the depth of the prismatic cell, 



when the latter has the same capacity as the real cell, as 1| to 5, 



then, taking the side of the hexagon as unity, we have for the same 



depth (viz. the longest side of the trapezium in the real cell) the 



25 VS 

 value ; and then (to three places of decimals): 



Area of the three rhombs, f a/2 

 „ „ „ six triangles, f V2 

 „ „ „ six sides of the equivalent 



prismatic cell, -^^ V 3 

 „ „ „ hexagonal base, f VS 

 The whole surface of the real cell, accordingly, 

 = (i)-f-(iii)-(ii) = 23-772; 



* For the most part instantaneously; but sometimes, when there are two 

 positions of nearly equal potential energy, the film "creeps" from the less to the 

 more advantageous of the two. 



t We must take into account the depth of the cell, or assume a value for it, 

 if we are to estimate the percentage saving of wax on the whole construction. 

 But (as Dr G. T. Bennett says) the whole saving is on the roof, and the height 

 of the house does not matter; the question rather is, what is saved on the 

 rhomboidal sloping roof compared with a flat one? If the short axis of the rhombs 

 be 2 units (the edge of the cube), then 3 rhombs have area 6 V2, the wall-saving 

 is 2 V2, while the flat hexagonal top is 4 Vs. So the actual saving is the diflFerence 

 between 4 V2 and 4 Vs — which looks much less negligible ! But it is only on 

 a small portion of the work. 



