TERMINOLOGY. . 116. 



For the derivation of these subordinate series, the first 

 process is applied to the scalene six-sided pyramids. 



Let ABXC, Fig. 47., represent the principal section of a 

 rhombohedron, 21BC that of a scalene six-sided pyramid 

 derived from it according to a certain m. 



If tangent planes are laid on the terminal edges 3B, &c. 

 of this pyramid ; these edges become the inclined diagonals 

 of the resulting rhombohedron. Suppose its axis = a' ; that 

 part of it which corresponds to the inclined diagonal 2tB 

 will be 



and 



ao, = |. a' = Ma + MQ= 3 m tJL 



6 



,_, 3m+ 1 3m+ 1 

 a_ 2 . -_.a .a, 



the side of the horizontal projection BQ being = I. 



If on the other side tangent planes be laid on the acute 

 terminal edges aC, &c. ; these terminal edges again will 

 become the inclined diagonals of the derived rhombohe- 

 dron. The axis of this rhombohedron being = a" ; that 

 part of it which corresponds to the inclined diagonal aC 

 willb 



ap = . a" = Ma MP = L!Lrzi a 



and 



a" = L?_rJ. a 



4 

 for the same horizontal projection. 



Hence 3 m + J- and 3 m "ll are the co- efficients, with 



4 4 



which a, the axis of R, or 2?. a, the axis of R + n must be 

 multiplied for obtaining members of the subordinate series. 

 When these co-efficients become powers of the number 2, 

 the rhombohedron produced is a member of the principal 

 series ; when they are not powers of the number 2, mem, 

 bers are produced belonging to a subordinate series, which 

 is determined by m. 



Designate the subordinate series by R 4- 



The quantity 3 m "Li. 2". a substituted for 2". a in the 



