Philosophical Transactions. 297 



remains but to express by algebraical equations the planes which 

 result from any assigned laws of decrement, for the different 

 primitive forms which occur in crystallography. 



To this effect the author assumes one of the angles of the pri- 

 mitive form, supposed, in the first case, a rhomboid, as the origin 

 of three co-ordinates, respectively parallel to its edges, and supposes 

 any secondaiy 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 this decrement will be such that the coefficients 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- 

 oridnates. 



To represent any face, the author encloses between parentheses 

 the reciprocal coefficients of the three co-ordinates of its equation, 

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

 three edges to form it, with semicolons between them. Tnis he 

 calls the symbol of that face. He then shews how tnincations on 

 all the different edges and angles of the primitive form are re- 

 presented in this notation, by one or more of the elements of 

 which the symbol consists becoming zero or negative, thus com- 

 prehending 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, ♦^^o form a perfect secondary crystal. This gives rise to co- 

 existent planes. In the rhomboid these coexistent planes are found 

 by simple permutations 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 coexistent planes which truncate the different angles 

 of the primitive form, as referred to that particular angle which 

 he assumes as the origin of the co-ordinates. It follows from 

 this analysis that, in this latter case, each of the elements of the 

 symbol must be combined with its excess over each of the remain- 

 ing two, to form a new symbol. This gives four symbols, includ- 

 ing the original one, each susceptible of six permutations, making 

 in all twenty-four faces. 



The author then proceeds to consider tlie cases of the irregular 

 tetrahedron and octohedron, the triangular prism and rhomboidal 

 dodecahedron, investigating in each case the symbols of the co- 

 existent planes, and illustrating his theory with examples taken from 



