126 On Crystallography. 



The smallest number of faces which the secondary cry- 

 stal can have is twelve. Then all the decrements consi- 

 dered two by two, setting out from one and the same ridgey 

 v. ill be inverse to each other. 



The simplest case is that in which the generator parallelo- 

 pipedon being a cube, we have n— i and //— l. On this 

 •hypothesis, the secondary solid is a dodecahedron with 

 rhombic planes all equal and similar, as we have explained 

 m the reasoning part of the work. 



16* We now proceed to the method of determining the 

 decrements upon the angles. But we mav previously re- 

 mark, that in this case the decreasing parts of the lamina?: 

 of superposition form angles alternately re-entering and 

 salient^ in Stteb a way, however, that all the ridges of mo- 

 lecule situated at the places of the salient angles are on 

 one and the same plan : we shall consequently designate 

 the series of these ridges by the name of lateral face. 



This being done, let us conceive that decrements in 

 breadth are produced by equal numbers of rows on all the 

 angles of the parallelopipedon, fig. 1 ; and let us take for 

 instance that which takes place upon the angle BCD. 

 Let Ckl be the mensurator triangle, in which C/e measures 

 the distance between the point C and the first lamina of 

 superposition, kl is regarded as being applied on the cor- 

 responding lateral face, the height of which it measures, 

 and C / coincides with the face of the secondary crystal, 

 produced by the decrement in question. 



Having traced the diagonals db,fh (fig. 2) on the 

 bases of the molecules, I draw ct perpendicular upon d b, 

 and x% perpendicular as well upon db as upOQ\fk, 



Let N be the number of rows subtracted. We shall 

 have Ck (6g..l) ssNXcf (fig. $}, and k I (fig. 1) = x z 

 (fig. 2); besides the angle Ckl (fig. l) will be equal to 

 that formed by the plane bdfk (fig. 2) with fgh. Now 

 these three quantities are regarded as being known, since 

 the form of the molecule is determined. Thus it will be 

 easy to find the angle kCl (tig. l) which measures the 

 inclination of the face produced by the decrement upon 

 the parallelogram ABCD. We shall conduct ourselves in 

 the same manner in order to calculate the effects of the de- 

 crements on the other angles. 



17. Let us now consider the hypothesis in which the 

 decrements which take place on the two angles DCG, 

 BCG would have such a connexion with that which acts 

 on the angle BCD, that the faces produced by these three 

 decrements would coincide on one and the same plane. 



Let 



