118 TERMINOLOGY. . 115. 



ably to the law of the series of rhombohedrons 

 (. 110.). 



The axes of the members of this series may be consider- 

 ed as products of the axes of II + n by m, that is to say, 

 as being equal to 2". m. a ; and since m and a are common 

 factors to them all, the axes are to each other in the ratio 

 of2(. 110.). 



If these values, namely 2 n . a instead of a, are substitut- 

 ed in the expressions . 55., the cosines of the edges are 

 obtained for any particular member of the series. 



. 115. LIMITS OF THE SERIES OF SCALENE SIX- 

 SIDED PYRAMIDS. 



The limits of the series of scalene six-sided pyra- 

 mids are, on the one side unequiangular twelve-sided 

 prisms of infinite axes, whose transverse sections 

 are determined by m, and on the other side plane 

 figures equal and similar to the horizontal projec- 

 tion. 



The axis of a scalene six-sided pyramid, which belongs 

 to a rhombohedron, whose axis is in a finite ratio to the 

 side of its horizontal projection, can never become infinite, 

 unless m itself should be infinite, a supposition, however, 

 which is excluded from the relations to be taken here into 

 consideration, in as much as such an m would give no 

 series, or rather a series, all the members of which are 

 equal to each other. Therefore the limits of the series of 

 these pyramids must arise from rhombohedrons, whose 

 axes are themselves infinite, or from the regular six-sided 

 prism (. 111.). 



Lay planes of intersection through the terminal points 

 of the lateral edges of a scalene six-sided pyramid derived 

 from a rhombohedron, whose axis is longer than the side of 

 its horizontal projection ; and thus detach the apices of 



