20 Mr. S. Sharpe on the Figure of the Cells of the Honeycomb. 



had, and shall therefore consider his measurement to have been 

 correct. 



By the fluctional theorem de maximis et minimis, the calcu- 

 lated angles appear to be wrong, and should be 109° 28' 16", 

 and 70° 31' 44?" ; which must be held to agree with the measure- 

 ment ; as no one would pretend to make that more accurate 

 than to the nearest minute, and the difference is much within 

 those limits. 



Kcenig's paper was read before the Academie des Sciences 

 at Paris in 1739, and the results are mentioned in their Trans- 

 actions for that year ; but as his working is not added, I can- 

 not compare mine with his to see where the error lies. 



As it cannot be held unimportant to show that bees build 

 their cells exactly in the form which, at length, by the dif- 

 ferential calculus, we find to be best ; and as I cannot expect 

 my assertion to be preferred to that of Kcenig, — I add the 

 working at length. 



1st. As of those figures which can be brought together 

 without leaving any interval, the hexagon is that which has 

 the greatest number of sides, it is clearly the one which needs 

 the least materials to inclose a given space. 



2ndly. If a range of hexagonal cells be met by a range 

 of similar cells, and no space be wasted, each cell must end 

 either with a three-sided pyramid, or with a plane at right an- 

 gles to the sides (i. e. a pyramid whose altitude = 0). 



Query. What must be the altitude of the pyramid, in order 

 that a cell of a given prism and given solid contents may be 

 constructed of the smallest quantity of materials? 



Let fig. 1. be any such regular three-sided pyramid on a 

 six-sided prism. 



Fig. 1. 



Fig. 2. 



Fig. 2. the same prism with a plain end at right angles to 

 the sides («'. e. the altitude of the pyramid = 0). 



Fig. 3. 



