Royal Society of London. 325 



Thursday, Nov. 25. — A paper was communicated by W. Whe- 

 well, Esq. F.R.S., on A general metliod of calculating the Angles 

 made by any Planes of Crystals, and the Laws according to which 

 they are formed. 



The object of this elaborate paper was to propose a new system 

 of notation for expressing the planes of a crystal, and their laws of 

 decrement, and to reduce the mathematical part of crystallography 

 to a few simple formulae of universal application. The author 

 proposes to represent each plane of a crystal by a symbol indica- 

 tive of the laws from which it results, which by varying only its 

 indices, may be made to represent any law, and by means of which, 

 and of the primary angles of the substance, a general formula 

 may be derived expressing the dihedral angle between any one 

 plane resulting from crystalline laws and any other. The angle 

 contained between any two edges of the derived crystal may also 

 be found in the same manner ; and conversely, having given the 

 plane, or dihedral angles of any crystal, and its primary form, the 

 laws of decrement according to which it is constituted may be 

 deduced by a direct and general process. 



The mathematical part of this paper depends on two formulae, 

 by 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 func- 

 tions of the coefficients of their equations supposed given. These 

 formulas being taken for granted, it remains to express by alge- 

 braical equations the planes which result from any assigned laws of 

 decrement for the different primitive forms. For this purpose the 

 author assumes one of the angles of the primitive form, supposed, 

 in the first case, a rhomboid, as the origin of three co-ordinates 

 respectively parallel to its edges, and supposes any secondary face 

 to arise from a decrement on this angle by the subtraction of any 

 number of molecules on each of its three edges. It is demon- 



