314 Rev. Mr Whewell's General Method of calculating 



He then shews how truncations on the edges and angles of 

 the primitive form are represented 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 oc- 

 cur in one uniform analysis. 



The law of symmetry in crystallography, requires that si- 

 milar angles and edges of the primitive form should be modi- 

 fied similarly, to produce a perfect secondary crystal. This 

 gives rise to co-existent planes. 



In the rhomboid, three co-existent planes are formed by 

 simple permutation of the elements of the symbol, one among 

 another. In the prism, such only must be permitted as re- 

 late to similar edges. 



In other primitive forms, such as the tetraedron, Mr 

 Whewell institutes a particular inquiry into the decrements 

 of the co-existent 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. In this 

 latter case it follows, from the analysis, that each of the ele- 

 ments 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, mak- 

 ing in all twenty-four faces. 



Mr Whewell then considers a variety of other cases, and 

 treats of the order in which the faces lie in a perfect crystal, 

 and the determination of such faces as are adjacent, or other- 

 wise. Lastly, he investigates the angles made by edges of the 

 secondary form. 



The following formulae may be used for calculating the 

 angles made by any secondary faces of a crystal, when the 

 law of its derivation from the primary is known ; and con- 

 versely, for determining the law of formation when the angles 

 of the secondary form are given. 



Let any solid angle, contained by three plane angles of the 

 primary form, be considered as the origin of our measure- 

 ment ; let x, y, z be the three edges, formed by the meeting 

 of the three planes. Let any secondary plane, cut off from 

 x, y, Xf lines of which the reciprocals are p, q, r, respectively. 



