1862.] 191 



It is the object of the present paper to show that not only the forms 

 of the pyramidal and rhombohedral systems, but also those of the pris- 

 matic, are as strictly isomorphous as those of the cubical system. 

 This is effected by demonstrating all the forms of these three systems 

 to be but partial developments of the cubical system. Consequently 

 every form can be indicated by the symbols of the cubical system. 

 In other words, instead of having distinct axes and parameters 

 for each system, and for every substance in that system, all are 

 referred to the rectangular axes and equal parameters of the cubical 

 system. 



The pyramidal system is regarded as a tritohedral development of 

 the cubical system, the faces so developed being all symmetrically 

 taken with respect to one of the cubical axes. Thus, adopting the 

 notation for the cubical system in the last edition of Phillips's 

 ' Mineralogy/ the two faces of the cube 001 and 001 will form the 

 basal pinacoids, while the remaining faces 1 0, 1 0, 1 0, and I 

 will give the direct square prism. 



The faces of the rhombic dodecahedron 110, fl 0, TTO, and 1 10 

 will give those of the inverse square prism. 



There will be two groups of square pyramids derived from the 

 four-faced cubes, the first in the development of the faces indicated 

 by the symbols 



, Qkh, Oh, Qkh 



, OAA, lOA, Oil; 

 the second by 



hOk, Qhk, HOk, Ohk 



AOl, Qhlc, htilc, 



the poles of both of these forms always lying in the zone 001, 100. 

 The remaining eight faces of the four-faced cube, when developed 

 symmetrically with respect to the cubic axes, viz. 



hkQ, MO, MO, ikO, AO, MO, MO, and hkQ, 



give the octagonal prism. 



The inverse square pyramids are derived from the tritohedral 

 development of the faces of the twenty-four-faced trapezohedron and 



