190 TERMINOLOGY. . 149- 



is changed either into an unequiangultir eight-sided prism, 

 or into a plane perpendicular to the axis, the situation of 

 the edges in 2. remains nevertheless unchanged, provided 

 in the case of the prism, the position still remains the 

 parallel one and m = m'. Ex. of the latter, (P + I) 3 (e) 

 and [(P 4- co) 3 ] (/) in pyramidal Garnet. Vol.11. Fig. 96. 



4. There exists in the pyramidal system, a series of sca- 

 lene eight-sided pyramids analogous to that of the scalene six- 

 sided pyramids in the rhombohedral system (. 145. iv. 4.). 

 Its members succeed each other in the following order: 

 ... (P + n) s , (P 4- n + I) 4 , (P 4- n +2) 3 , (P + n + I) 5 ... 

 in which the consecutive members assume a diagonal posi- 

 tion towards each other. Ex. (P) 4 (.r) and (P + I) 3 (e) in 

 pyramidal Garnet. Vol. II. Fig. 96. The edges of combi- 

 nation between the faces of every two subsequent pyramids 

 are parallel to each other, to the more obtuse terminal edges 

 of the lower, and to the more acute terminal edges of the 

 higher member in the series. The demonstration of this 

 property depends upon the same suppositions as in the six- 

 sided pyramids. 



Let A'B, Fig. 70., represent the more obtuse terminal 

 edges of (P 4- n) m ; we find the algebraic expression of 



n 



. i/T> , r in. 2 5 . a 

 sin A'BM = 



a 2 + 2) 



If in the same way we suppose A'C, Fig. 69., to be the 

 more acute terminal edge of (P 4- n') !tl/ ; a similar algebraic 

 expression will give 



(m' 4- 1) 2'. a 



sinA'CM= ililJL.^ 



l(m' + I) 2 . 2< a 2 -f 4] 

 These two expressions become equal, if in the first we 

 substitute 5 for m, and in the second n 4- 1 for n', and 4 

 for m'. They again become equal if in the first we suppose 

 m = 4 ; in the second n' = n + 1 and m' = 3 : and again 

 for m = 3 in the first, and n' = n 1, and m' = 5 in the 

 second expression. For the rest, the remarks of . 145. iv. 5. 

 find here equally their full application. 



