120 TERMINOLOGY. . 115. 



remains unaltered ; and since the angles of the transverse 

 section are solely dependent upon m, the unequiangular 

 twelve-sided prism will have the same transverse section as 

 any other member of that series, whose limit it represents. 

 As to the opposite limit, it is evident, that if the height 

 of the central part, = i of the axis of the rhombohedron, 

 disappears, the axis of this rhombohedron itself must also 

 be = ; and consequently that the rhombohedron must be 

 a plane figure, equal and similar to the horizontal projec- 

 tion. Now the axis of the pyramid sought is m. = ; 

 and the pyramid therefore likewise a plane figure, equal 

 and similar to the horizontal projection. 



The designation of the series between its limits is 



R _ co ... (P + n) ... (P + cc ). 

 If an unequiangular twelve-sided prism, from its original 

 position, is transferred into another different from it for 

 60 or 180, its faces and edges resume exactly the situa- 

 tion they had before. Hence there is only one position 

 existing for these forms : in which property it agrees with 

 the regular six-sided prisms, the limit of the series of rhom- 

 bohedrons, from which it derives its origin.* 



If in the algebraic expressions mentioned in the last 

 paragraph, n is made = co ; the angles are obtained for the 

 transverse section of the unequiangular twelve-sided prism, 

 being the limit of the respective series of the scalene six- 

 sided pyramid. Thus we find 



m 2 + 6 m 



* Another regular six-sided prism, which in every respect, 

 but the position, agrees with the former, will be considered 

 in . 1 18. This form, however, is in no immediate connexion 

 with the scalene six-sided pyramids ; and consequently no 

 unequiangular twelve-sided prism can be considered in, or 

 referred to, a position analogous to that regular six-sided 

 one, although the angles of their transverse sections should 

 seem to indicate a similar position. 



