178 On the Analysis of the Rhotnbohedral System. 



The equation (2) is given by Professor Miller in his i Trea- 

 tise on Crystallography/ 1839, in the form 



tan PO tan XO cos XOP= ^-TTT' 



h + k + l 



and in the equivalent form 



9/, — h — 1 

 2 tan PO cot OA cos XOP = 7 , 



the latter being the same as (2), with the sole difference that 

 tan OQ is replaced by its value given by equation (A). The 

 form in which it is given by Professor Miller does not, how- 

 ever, bring out so prominently the simplicity and directness 

 of the relation existing between the quantities involved in the 

 equation and those given by observation. 



As an illustration of the utility of equations (2) and (2'), 

 let us take the determination of a scalenohedron on a mineral 



(such as calcite) whose elements are known. Measurement of 

 two of the angles between adjacent faces suffices for the deter- 

 mination. If PP, and PP„ are the two angles measured, we 



know the three sides of the triangle aPa 7/ ; and the angle 



rrr 



P«6 = 6Q= — — OQ is the first quantity deduced from the 



measurements. Equation (2) then gives a simple equation in 

 terms of the indices A, k, l. If PP, or PP ;/ be given with the 

 angle of the middle edge of the scalenohedron, we know the 

 sides of the triangles aPa or a t ^a r In the first case OQ is 

 determined as before, in the second OP; and we must employ 

 (2) or (2 f ) accordingly. 



To complete the analysis, I need only point out that the 

 relations connecting the indices of dirhombohedral forms can 

 be most simply obtained by aid of the equations connecting 

 the indices of a face with those of the zone in which it lies. 

 Thus E, the face of the dirhombohedral form corresponding to 

 P, lies in the zones [OP] and [&/P//], whence its indices can 

 be at once obtained, and all the geometrical relations connect- 

 ing it with P can be proved. Professor Maskelyne has, I 

 believe, already given this method of deducing the indices of 

 the dirhombohedral form in his lectures at Oxford. 



