. 143. OF COMBINATIONS. 155 



in it, there is no possible case, in which more than two data 

 are required, each of these data consisting in the situation 

 of a straight line upon one of the faces of that form. 



This method of determining the relations of simple 

 forms contained in a combination, is evidently founded 

 upon the knowledge of the series produced by these forms, 

 which have been explained above : by these it acquires a 

 perfect generality, because in every stage the same rela- 

 tions exist among the members. 



The developement itself may be effected either analyti- 

 cally or synthetically. The synthetical method speaks more 

 plainly to the eye, and is therefore particularly recom- 

 mended to beginners. The analytical method is more easy, 

 elegant, and general. Several examples of the synthetical 

 method are contained in the course of this work ; and since 

 it would far exceed its limits to treat of these methods at 

 large, I shall only subjoin a short sketch of the process of 

 the analytical method. 



Let ABC, A'B'C', Fig. 49., represent the faces of two 

 forms of the rhombohedral system, for instance, of two sca- 

 lene six-sided pyramids, whose horizontal projection is the 

 same, and which are placed in a parallel position. . The 

 lines CB, C'B' will intersect each other in the point G, and 

 six points situated like G will be common to both the forms. 

 The points G, G &c. are situated in a horizontal plane, per. 

 pendicular to the axis AX in M, the centre of the form ; 

 they are constant in all forms of the rhombohedral sys- 

 tem ; for though the situation of the points C, C' and 

 B, B' may vary, yet this never can have any influence 

 upon their intersection in G. In the other systems of 

 crystallisation, the situation of the points G, G &c. is 

 not invariable ; but it may easily be shewn, that this si- 

 tuation depends upon the diagonals of the bases of the 

 forms combined. 



The acute terminal edges AC, A'C', intersect each other 

 in the points G', G', &c., which points, therefore, are likewise 

 common to both the forms. The situation of these points 

 is variable, and depends upon the relations of the axes, be- 



