326 Proceedings of the 



strated, first, that the equation of the plane arising from this decre- 

 ment will be such that the co-efficients of the three co-ordinates 

 in it (when reduced to its simplest form) will be the reciprocals of 

 the numbers 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 co-ordinates ; 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 

 passing through the origin of the co-ordinates. 



In order to represent any face, Mr. Whewell encloses between 

 parathenses the reciprocal co-efficients of the three co-ordinates of 

 its equation, with semi-colons between them. He then shews how 

 truncations on the edges and angles of the primitive form are repre- 

 sented in this notation, 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 

 angles and edges of the primitive form should be modified simi- 

 larly, to produce a perfect secondary crystal. This gives rise to 

 co-existent planes. 



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

 permutation 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, such as the tetraedron, Mr. W. institutes 

 a particular inquiry into the decrements of the co-existent planes 

 which truncate the difierent angles of the primitive form, as re- 

 ferred to that particular angle which he assumes as the origin of 

 the co-ordinates. In this latter case, it follows from the analysis, 

 that each of the elements of the symbol must be combined with its 

 excess over each of the remaining two to form a new symbol. 

 This gives four symbols, each susceptible of six permutations, 

 making in all twenty-four faces. 



Mr. W, then considers a variety of other cases, and treats of the 

 order in which the faces lie in a perfect crystal, and the determina- 



