On the Mathematics of Bees^ Cells, 



229 



«ets ot' apexes are represented by the two symbols • and O? 

 and lie at equal distances A from the plane of the paper on 

 opposite sides of it. All points of meeting not so marked lie 

 in the plane of the paper. Two sets of hexaoons are indicated, 

 one bv continuous and the other by dotted lines. They are 

 the ])Vojections of the two sets of prisms. The rhombuses 

 into which the figure is divided are projections of the rhom- 

 buses which meet at the several apexes ; each rhombus-edge 

 being in the plane of one of the prism-faces. The plane of 

 one set of apexes is at a distance '2h from the plane of the 

 other set. 



The projected length of a rhombus-edge is s, and its actual 

 length \/(>-h/r). 



The long diagonal of a rhombus lies in the plane of the 

 paper, and is .? v/ 3 ; the shorter diagonal is V [s^ -\-W). If 

 i) denote an acute angle of a rhombus we have 



whence 



cos^t^- 2 \/{s^ + li'y 



cos Q = 



s' -2h' 



If (j) denote the acute angle which a prism-edge makes with 

 the rhombus-edge that it meets, we have 



Changes in the A^alue of h will not affect the volume of a 

 cell, but they will affect the quantity of wax required for 

 building the celL If h were zero the prism-edges would all 

 terminate in the plane of the paper, which would also 

 contain the rhombuses. As compared with this standard 

 of reference, the area of a prism-face is less by ^ sh, and the 

 area of a rhombus is greater hy ^s \/o's/(s'^-\-4:Jr) — ^s^\/ 3. 

 To give pro])er weights to these two items on opposite sides 

 of the account, we must know the ratio of the number of 

 prism-walls to the number of rhombuses. Each cell has twice 

 as many prism-sides as rhombuses ; and in the comb, except 

 at the outside, each rhombus, as well as each prism-wall, is 

 common to two cells. At the outside, the walls exposed also 

 comprise twice as many prism- walls as rhombuses. We must 

 therefore reckon two prism-faces to one rhombus, and the 

 net saving in area is 



sh-is\/3{s/{s'-]-U')-s\, 



