. 14)5. OF COMBINATIONS. 171 



If in the hypothesis of (P + n) 5 . (P + n + I) 3 , we sub- 

 stitute in sin. &BQ, the number 5 instead of m, and in 

 sin. SCP the expression n + 1 for n', and 3 for m' ; we 

 obtain 



4.2 2 ".a 2 +36)' 

 two values which are equal. 



If the above mentioned relations take place among the 

 consecutive members of the series, the parallelism of the 

 edges of combination always must follow. But these re- 

 lations are not a necessary consequence of that parallelism, 

 and many pyramids really exist and produce in their regular 

 positions, parallel edges of this kind ; and yet they are not 

 derived from rhombohedrons of the same series at all, or ac- 

 cording to other values of m, than those of 2, 3, and 5. This is 

 the case in the combinations of (P I) 3 (a) and (f P I) 3 (6) 

 in rhombohedral Ruby-blende. Vol. II. Fig. 126. In the 

 above mentioned series, one datum has been assumed, 

 upon which the rest is dependent. If this be changed, the 

 consequences also must be altered, without, however, in 

 the least affecting the parallelism of the edges of combi- 

 nation. Supposing the pyramids to be derived from the 

 members of the principal series, we may argue with per- 

 fect security, from the parallelism of the edges to the va- 

 lues of m, and determine the one whenever we know the 

 other. Restrictions of this kind frequently occur in general 

 solutions of crystallographic problems. 



v. (P + n) m . P -f n'. 



1. Let m be = 5, n' = n + 3. We have the combina- 

 tion (P -|- n) 5 . P + n + 3. The faces of the scalene pyra- 

 mid meeting in its obtuse terminal edges, appear in the 

 place of the alternating terminal edges of the isosceles one. 

 The edges of combination are parallel to each other, and to 

 the above mentioned terminal edges of the two forms. For 

 on these suppositions the inclination of the terminal edges of 



