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planes, which truncate the different angles of the primitive form, as 

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

 the coordinates. It follows from this example, that in this latter 

 case 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 (including the original one), each susceptible of 

 six permutations, making in all 24 faces. 



The author then proceeds to consider the cases of the irregular 

 tetrahedron and octohedron, the triangular prism, and rhomb dode- 

 cahedron, investigating in each case the symbols of the co-existent 

 planes, and illustrating his theory with examples taken from the 

 crystalline forms of zircon, sulphur, and other minerals. He next 

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

 determination of such faces as are adjacent or otherwise. To this 

 end, he conceives an ellipsoid inscribed within the crystal, having for 

 its three axes the three most remarkable lines in the primitive form, 

 and by means of the well-known equation of the second degree repre- 

 senting such an ellipsoid, combined with the equation of any pro- 

 posed, he deduces the longitude and latitude, on the surface of the 

 ellipsoid, of the point at which it would be touched by a plane paral- 

 lel to such face. The results are included in general and explicit 

 formulae, by whose application, in any proposed case, the sequence 

 and arrangement of the faces in the perfect crystal are readily dis- 

 covered. 



The angles made by edges of the secondary form are next investi- 

 gated ; after which the author, having recapitulated his results, takes 

 occasion to refer to a paper by Mr. Levy, who had previously, but 

 unknown to Mr. Whewell, employed the representation of a secon- 

 dary plane, by its equation referred to the three principal edges of 

 the primitive form, but only in a particular case ; whereas the inves- 

 tigation and notation in the present paper are absolutely general. 



In the course of this paper, Mr. Whewell instances the application 

 of his analysis to the solution of the following problems : 



Knowing the dihedral angles of the secondary rhomboid, to find 

 the symbol of its faces, or their laws of decrement. 



To find what laws of decrement give a secondary rhomboid similar 

 to the primary one. 



Knowing the lateral angles made by the planes of any bipyramidal 

 dodecahedron, to find the symbols. 



Knowing the angles made by any plane, with two primary planes 

 to find its symbol. 



To find what laws give prisms parallel to the axis of the rhom- 

 boid. 



To find the symbol of a plane which truncates the edge of any 

 secondary rhomboid. 



