of Crystalline Combinations. 395 
angles. The radii of the circles passing through the original 
4 
and the derived angles (MB and MS, Fig. 2.), are 6 and ss = 2 
x 
The axis of pRm is ch pa. 
H. If b be the radius of the circle circumscribing the base of R, 
the base of pRm is a hexagon with alternately equal angles, 
and the radii of the circle passing through the original and 
the derived angles (OB and OS, Fig. 1.), are 6 and a 
I, The acuter and obtuser edges of pRm make with the axis 
angles of which the tangents are respectively 
2 b 2 b 
3m—1 pa “"° 3m-+1 pa’ 
The demonstrations of the preceding theorems will be given 
in another place. To complete the subject, it will also be requisite 
to give methods of transforming the symbols, which are employed 
in other systems of notation hitherto proposed, (those of Hauy, 
Weiss, Mohs, &c.) into this system. This will be done by 
means of certain formula, which we may call formule of trans- 
formation. 
At present our object is to shew how, by means of the the- 
orems above enunciated, with the additional assistance of a few 
subsidiary theorems derived from them, we may reason from the 
parallelisms and other properties of crystalline forms, so as to 
obtain the indices which belong to those forms; and the rules 
for drawing these inferences may be called Canons or DeRtva- 
TION. 
Our object, therefore, will be to divide the combinations of 
crystalline faces into classes, so that each class may offer some 
peculiarity in the position of its edges and faces, by which we may 
recognize the laws from which they result. These peculiarities 
Vol. WI. Part I. $3 E 
