On Crystallography, 125 



(fig. 2), and oilier cs. Thus it will be easy to determine 

 the ancrle which the face produced by the decrement forms 

 with BCGH (fig. 1 ) 



13. It may happen that the two decrements which act 

 on both sides of the ridge BC have such a connexion 

 with each other that the two faces which will result from 

 them will coincide upon one and the same plane, so that 

 the side oh of the triangle oih is upon the direction of 

 the side ou which belongs to the triangle upo, as we see 

 in fig. 3. . To prove this, we may remark, that in this 

 case the triangles upo, oik are similar, as well on ac- 

 count of the equality of the angles op u, hi o, and the pa- 

 rallelism of the sides op, i h, as on account of the coinci- 

 dence of the sides ou, ho upon one and the same direction. 



Therefore pu : op : : oi : ill. 



Or rather cs (fig. 2) : n x cr : :n' x cs : cr. 



Which gives n' = — . 



That is to say, the case in question will happen every 

 time that the decrements which take place on proceeding 

 from BC towards GU are in the inverse ratio to those 

 which take place in going from BC towards AD, or, what 

 comes to the same thing, at all times when there is on one 

 side a decrement in height equal to that which shall be on 

 the opposite side. We may easily conceive that the two 

 faces will be still on one and the same plane in the peculiar 

 case of a decrement by one row on both sides. 



14. Hence we may conclude, that in all circumstances- 

 similar to those which have been cited, we may make abs- 

 traction of one of the two decrements, by considering the 

 face which results from it as the continuation of that which 

 arises from the other decrement. 



We see what would have been necessary for determining 

 in a similar way the incidences of the faces produced by 

 the other decrements upon the analogous faces of the 

 generator parallelopipedon. 



15. The greatest number of faces which the secondary 

 solid can have, is twenty-four, since the generator parallelo- 

 pipedon has twelve ridges, each of which is the line of 

 departure of two decrements which act in an opposite direc- 

 tion. These faces will all be triangles, or some triangles and 

 others trapeziums, according as the generator paralielopipe- ■' 

 don will be found more elongated in one. direction than in 

 the other, or as the decrements which will take place pa- 

 rallel to certain edges will follow a more rapid course than 

 those which would act parallel to other edges. 



The 



