148. OF COMBINATIONS. 181 



and hence m' = 3. The pyramid accordingly will be per- 

 fectly designated by the crystallographic sign (P) 3 . 



The more obtuse pyramid produces with (P) 3 horizontal 

 edges of combination, m is therefore = in 7 = 3 (. 145. 

 iv. 2.); and (P 4- n iv ) m = (P 4- n")3. 



But the faces of R 1 appear with parallel edges of 

 combination, in the place of the more acute terminal edges 

 of this pyramid. If, therefore, a' be the axis of the rhom- 

 bohedron to which the pyramid belongs, whilst a is the 

 axis of R, we have from . 145. ii. 4., 



3.3 1 a , _ _2_ a . 

 ~~6"~ 3 ~2~' 



from which follows 



a'= i. a = 2-2. a; 

 n iv is therefore = 2, and (P + 11^)3 _ (p __ 2)3. 



The edges of combination between the pyramid just now 

 determined and the rhombohedron R -f- n" are parallel to 

 the more obtuse terminal edges of the former, and to the 

 terminal or lateral edges of the latter. The two forms 

 rank under the head of . 145. ii. 7- 



Let a' be the a^sis of the rhombohedron R -f n". We 

 have 



and accordingly, 



n 11 is therefore = ; but the rhombohedron belongs to the 

 first subordinate series, and R + n 11 is = R. 



If now we write down the signs of the forms as we have 

 found them in the developement in their regular order ; 

 we obtain the crystallographic designation of the combina- 

 tion, disposed according to the different angles which per- 

 pendicular lines drawn upon the faces produce with the axis, 



R 1. (P 2)3. R. | R. R + l. (P) 3 . R + 03. 

 a I c d e f g 



The 60th figure represents a di-rhombohedral combiua- 



