1885.] Geometrical Construction of the Bees Cell. 



253 



IV. " On the Geometrical Construction of the Cell of the 

 Honey Bee." By Henry Hennessy, F.R.S., Professor of 

 Applied Mathematics in the Royal College of Science, 

 Dublin. Received October 20, 1885. 



The well-known problem of the bee's cell occupied the attention of 

 eminent mathematicians early in the last century, and it is still pre- 

 sented as an interesting example of geometrical maxima and minima. 

 In 1743 Maclaurin communicated to the Royal Society a solution of 

 the question, which appears in the " Philosophical Transactions,"* and 

 it seems that the comparison between the mathematical results and 

 the actual cells was effected by angular measurements. Long since 

 a simple method occurred to me for the construction of the figure 

 without employing angles, and as I have not been able to find it in 

 any publication, I venture to submit it in this short paper. 



A structure has a regular hexagon for its orthogonal cross section, 

 and is terminated by three lozenges which meet in a trihedral angle ; 

 required the relation between the side of one of these lozenges and the 

 side of the regular hexagon forming the cross section of the prism so 

 as to give the smallest surface to the structure. The long diagonal 

 of one of these lozenges is equal to the side of the equilateral triangle 

 inscribed in the hexagon ; if we call this E and the short diagonal e, 

 then — 



E = </3h and e— ^W+4aP, 



where h is the side of the hexagon, and x the difference between the 

 parallel sides of one of the six faces of the prism. The area of a 



lozenge is therefore — ^ ~^^ x ? and that of the face of prism 



Li 



h(2l-x) 

 2 ' 



Eor u, the total surface of the structure, we have 



U =Y^ * / 3(fc? + 4a>»)+ 2(2l-x) ], 



and hence — =3/i f~ 2x '/ S__ # 



dx LVW + 4<x 2 J 



This, equaled to zero, gives 



A side, s, of the lozenge is manifestly x 2 , hence 



s=3x. 



* Vol. 42. 



