Mr. J. W. L. Glaisher on the Form of the Celts of Bees. 113 



the prismatic cell with a plane hexagonal floor as 25 -f V§ to 

 28, so that the wax saved is about the one fifth part of that re- 

 quired for the plane floor. Lhuillier merely proves the above value 

 for the ratio • but it is interesting to have the amount of wax 

 required for the different portions of the cells ready for compa- 

 rison. Taking, therefore, a side of the hexagonal section as 

 unity, and assuming with Lhuillier (after Maraldi's observations) 

 that the ratio of the radius of the inscribed circle of the hexagon 

 to the depth of the prismatic cell of the same capacity as the 

 real cell is as 14 to 5, we have for this depth (viz. the longest side 



of the trapeziums in the real cell) the value — — — > anc * * nen *° 



three places of decimals — 



9 v^2 

 Area of the three rhombs . . . = — — = 3*182, (i) 



„ six triangles . . . . = = 1*061, (ii) 



„ six sides of equivalent"! 25^/3 __ „„-. /•••% 



prismatic cdl . ./ = 2 =21-651, (i„) 



q /q 



„ hexagonal base . . . = — - — = 2*598, (iv) 



and the surface of the real cell = (i) -f (iii) — (ii) = 23*772, while 

 that of the prismatic cell with a flat hexagonal bottom = (iii) + (iv) 

 = 24*249 ; so that the saving is *477, and the ratio is as stated 

 above. 



It forms no part of my present purpose to give a detailed ac- 

 count of the form of Lhuillier' s minimum minimorum cell ; suffice 

 it to say that he extends Reaumur's problem, and proceeds to 

 inquire what must be the proportion of the depth of the cell to 

 the width of its mouth that it may require the minimum of wax 

 (the shape of the cell, viz. the inclination of the rhombs to the 

 trapeziums and to one another, being supposed as in the bee- 

 cell). A few remarks on this paper will be made further on. 



Lhuillier refers to Lambert [Beytrdge zum Gebrauche der 

 Mathematik, t. iii. p. 387 et seq. Berlin, 1772) * but I find on 

 reference there is nothing that very closely relates to the subject 

 of this paper. Lambert's remarks occur in the course of an 

 essay on the Art of Building* and he prefaces chap. v. (on mi- 

 nima in roofs) with the statement that if it were usual to build 

 hexagonal houses we should obtain our model from the bees. 

 In point of fact hexagonal summer-houses, sentry-boxes, towers, 

 &c. are not very uncommon ; so that he investigates the condi- 

 tion of roofs of minimum surface both in this case and when the 

 base is square. As the result of the former problem he obtains 



