A general method of calculating the angles made by any planes of crystals, and the laws according to which they are formed

The author, after stating the inconsistencies, inelegancies, and imperfections of the received notation for expressing the planes of a crystal, and the laws of decrement by which they arise, and of the usual methods of calculating their angles, explains the object of the present paper, which is to propose a system exempt from these inconveniencies, and adapted to reduce the mathematical portion of crystallography to a small number of simple formulae, of universal application. According to the method here followed, each plane of a crystal is represented by a symbol indicative of the laws from which it results, which, by varying only its indices, may be made to represent any law whatever; and by means of these indices, and of the primary angles of the substance, we may derive a general formula expressing the dihedral angle contained betweenany one plane resulting from crystalline laws,and other . In the same manner we can find the angle contained between any two edges of the derived crystal. Conversely, having given the plane, or dihedral angles of any crystal, and its primary form, we can, by a direct and general process, deduce the laws of decrement according to which it is constituted. The purely mathematical part of this paper depends on two formulæ, demonstrated by the author elsewhere and here assumed as known; by means of one of which the dihedral angle included between any two planes can be calculated, when the equations of both planes are given; and by the other, the plane angle included between any two given right lines can in like manner be expressed by assigned functions of the coefficients of their equations, supposed given. These formulæ being taken for granted, nothing remains but to express by algebraical equations the planes which result from any assigned laws of decrement, for the different primitive forms which occur in crystallography.