VOL. XLII^] PHILOSOPHICAL TRANSACTIONS. 711 



point e in the apex of the figure, so that 3 rhombuses of this kind, with 6 

 trapezia, may complete the figure of the cell. Let o be the centre of the 

 hexagon, of which ok and kb are adjoining sides ; join cb and ko, intersect- 

 ing it in A ; and, because cob is equal to ckb, and ke equal to oe, the solid 

 EBCK is equal to the solid cbco ; from which it is obvious, that the solid con- 

 tent of the cell will be the same wherever the point e is taken in the right line 

 KN, the points c, k, b, g, n, and m, being given. We are therefore to inquire 

 where the point e is to be taken in kn, so that the area of the rhombus cebc, 

 together with that of the 2 trapezia cgne, enmb, may form the least superficies. 

 Because ec is perpendicular to bc in a, the area of the rhombus is ae X BC, 

 that of the trapezia cgne, enmb, is cg + en X kg ; these, added to the 

 rhombus, amount to ae X bc + 2kn X kc — ke X kc ; and because 2kn 

 X KC is invariable, we are to inquire, when ae X bc — ke X kc is a 

 minimum ? 



Suppose the point l to be so taken on kn, that kl may be to al as kc is to 

 bc. From the centre a describe, in the plane akb with the radius ae, an arc 

 of a circle ek meeting al, produced if necessary, in r; let ev be perpendicular to 

 AR in V, and kh be perpendicular to the same in h ; then the triangles lev, 

 LKH, LAK, being similar, we have lv : le :: lh : lk :: lk : la :: (by the sup- 

 position last made) kc : bc Hence, when e is between l and n, we have 

 LH -f LV (= vh) : lk + le (= ke) :: kc : bc; and when e is between Kand 

 L, we have lh — lt (= vh) : lk — le (= ke) :: kc : bc ; that is, in both 

 cases we have ke X kc = vh X bc ; and cons equently ae X bc — ke X 



kc = AE X BC — VH X BC = AE — VH X BC = AR — VH X BC = AH + VR 



X BC ; which, because ah and bc do not vary, is evidently least when vr 

 vanishes, that is, when e is on l. Therefore clbI is the rhombus of the most 

 advantageous form in respect of frugality, when kl is to al as kc is to bc. 

 This is the same method by which we have elsewhere determined the maxima 

 and minima, in the resolution^of several problems that have usually been 

 treated in a more abstruse manner. See Treatise of Fluxions, Art. 572, &c. 



Now because ok is bisected in a, kc" = ok'^ = 4ak^; and ac^ = 3ak% or 

 BC = 2ac = 2^/3 X ak; consequently kc : bc :; 2ak : 2/3 X ak :: ] : \/3, 

 and KL : al :: (kc : bc) :: 1 : ^3, or al : ak :: y 3 : ^2 ; and (because ak : 

 AC :: 1 : v/3) AL : AC :: 1 : ^^2 ; that is, the angle cla is that, whose tangent 

 is to the radius as \/2 is to 1, or as 14142135 to 10000000; and therefore is 

 of 54° 44' 8", and consequently the angle of the rhombus of the best form is 

 that of 109° 28' ^^"• 



By this solution it is further easy to estimate, what their savings may amount 

 to on this article, in consequence of this construction. Had they made the 



