434 PROFESSOR MILLER, ON THE POSITION OF THE AXES OF 



face meets the surface of the sphere, will be called the pole of that 

 face. The measurements express the angles between the perpendiculars 

 to the faces, or the supplements to the angles between the faces them- 

 selves. This method of representing crystalline forms appears to have 

 been first employed by Neumann, in his Beitrage zur Krystallonomie, 

 and afterwards by Grassmann and Uhde. It has the advantage of ex- 

 hibiting all the faces of a crystal without confusion in one figure, 

 each zone being distinguished by a great circle drawn through the 

 poles of the faces composing it, and also of allowing all the requisite 

 calculations to be performed by spherical trigonometry applied to the 

 equations 



T cos PX = T cos PY = 7 cos PZ, 

 h k I 



or to formulse deduced therefrom, X, Y, Z being the points in which 

 radii parallel to the axes of the crystal meet the surface of the sphere, 

 and P the pole of the face {h; k\ l), which is parallel to the plane 



h- + k\-^ I- = 0. 

 a b c 



ad, /3/3', ^f, ^^' will be used to denote the extremities of diameters 

 drawn parallel to the optic axes, and the two axes of elasticity which 

 are perpendicular to YY' . In Figs. 5, 6, 7, 8 the faces are denoted 

 by the same letters as in the treatises of Mohs and Naumann. The in- 

 clinations of the faces of crystal (1) and (2) are deduced from a mean 

 of the best measurements of thirty or forty crystals, and are probably 

 within 1' of the truth. 



The chemical notation and atomic weights are those employed by 

 Dr Turner, in the fifth edition of his Elements of Chemistry. 



