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suiting from crystalline laws, and any other. In the same manner 

 we can find the angle contained between any two edges of the de- 

 rived crystal. Conversely, having given the plane, or dihedral an- 

 gles of any crystal, and its primary form, we can, by a direct and 

 general process, deduce the laws of decrement according to which it 

 is constituted. 



The purely mathematical part of this paper depends on two formula?, 

 demonstrated by the author elsewhere and here assumed as known ; 

 by means of one of which the dihedral angle included between any two 

 planes can be calculated, when the equations of both planes are given ; 

 and by the other, the plane angle included between any two given 

 right lines can in like manner be expressed by assigned functions of 

 the coefficients of their equations, supposed given. These formulae 

 being taken for granted, nothing remains but to express by algebra- 

 ical equations the planes which result from any assigned laws of 

 decrement, for the different primitive forms which occur in crystal- 

 lography. 



To this effect, the author assumes one of the angles of the primi- 

 tive form, supposed, in the first case, a rhomboid, as the origin of 

 three coordinates, respectively parallel to its edges, and supposes 

 any secondary face to arise from a decrement on this angle, by the 

 subtraction of any number of molecules on each of the three edges. 

 It is demonstrated first, that the equation of the plane arising from 

 tlu's decrement will be such, that the coefficients of the three co- 

 ordinates in it (when reduced to its simplest form,) will be the re- 

 ciprocals of the number of molecules subtracted on the edges to which 

 they correspond. If the constant part of this equation be zero, the 

 face will pass through the origin of the coordinates ; if not, a face 

 parallel to it may be conceived passing through such origin, and will 

 have the same angles of incidence, &c. on all the other faces of the 

 crystal ; so that all our reasonings may be confined to planes pass- 

 ing through the origin of the coordinates. 



To represent any face, the author incloses between parentheses 

 the reciprocal co-efficients of the three coordinates of its equation, 

 or rather of the numbers of molecules subtracted on each of the three 

 edges to form it, with semicolons between : this he calls the sym- 

 bol of that face. He then shows how truncations on all the different 

 edges and angles of the primitive form are represented in this no- 

 tation, by one or more of the elements of which the symbol consists 

 becoming zero or negative, thus comprehending all cases which can 

 occur in one uniform analysis. 



The law of symmetry in crystallography requires that similar an- 

 gles and edges of the primitive form should be modified similarly to 

 form a perfect secondary crystal. This gives rise to co-existent planes. 

 In the rhomboid, these co-existent planes are found by simple per- 

 mutation of the elements of the symbol one among another. In the 

 prism, such only must be permuted as relate to similar edges. In 

 other primitive forms, as for example in the tetrahedron, the author 

 institutes a particular inquiry into the decrements of co-existent 



