. 145. OF COMBINATIONS. 165 



a = f . a', or a' = |. a = f . 2. a ; 



and R + n = f R + n', or that member of the first subor- 

 dinate series, which belongs to R + n'. 



Let now m' be = 5 ; we find that 



a = f . a', or a' = |. a = |. 2 1 . a, 



and R + n = | R + n' + 1, or that member of the second 

 subordinate series, which belongs to R 4- n' + 1. 



Each of the two co-efficients, f and that immediately fol- 

 lowing |, determines a particular subordinate series, which 

 may be distinguished by the name of the first and the second. 

 In itself it is quite arbitrary which of their members are 

 considered to be in the nearest relation to members of the 

 principal series. But it is very useful to fix upon a cer- 

 tain member, and this has been done here by supposing, 

 that, when the axis of R + n of the principal series is = 2 n . a, 

 that of R + n of the first subordinate series is f . 2". a, 

 and that of R + n of the second subordinate series |. 2 n . a. 

 Members of these dimensions are said to belong together, 

 or to be co-ordinate (. 116.). 



5. Let n' be = n 2 and m' = 5. The sign of the 

 combination will be R + n. (P + n 2) s . Under these 

 circumstances, the forms are in a parallel position. The 

 faces of the rhombohedron appear in the place of the more 

 obtuse terminal edges of the pyramid. The edges of combi- 

 nation are parallel with each other, with the above mentioned 

 terminal edges of the pyramid, and with the inclined dia- 

 gonals of the rhombohedron. Ex. R + 2 (m) and (P) 5 (y) 

 in rhombohedral Lime-haloide. Vol. II. Fig. 116. We 

 may infer inversely from this situation of the edges, that 

 the above mentioned relations really take place. 



For, making again use of Fig. 47., &B will represent the 

 obtuse terminal edge of the pyramid, but at the same 

 time also the inclined diagonal of that rhombohedron, 

 whose faces touch the more obtuse terminal edges of the 

 pyramid, their horizontal projections always being suppos- 

 ed equal. Hence, for the pyramid, we have 



ad = ^LtJ. a, 



