the Angles formed by any Planes of Crystals. 313 



present 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 result- 

 ing from crystalline laws, and any other. The angle contain- 

 ed between any two edges of the derived crystals, 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 consti- 

 tuted may be deduced by a direct and general process. 



The mathematical part of this paper depends on two for- 

 mulae, 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 ex- 

 pressed by assigned functions of the co-efficient of the equa- 

 tion supposed given. These formulas being taken for grant- 

 ed, it remains to express, by algebraical equations, the planes 

 which result from any assigned laws of decrement for the dif- 

 ferent 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 respective- 

 ly, 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 

 demonstrated, first, that the equation of the plane arising from 

 this decrement will be such, that the co-efficients of the three 

 co-ordinates in it (when reduced to its simplest form) will be 

 the reciprocals of the numbers of molecules subtracted on the 

 edges to which they correspond. 



If the constant part of this equation be zero, the faces will 

 pass through the origin of the co-ordinates ; if not, a face 

 parallel to it may be conceived passing through such origin, 

 and will have the same angles of incidence, &c. on all the 

 other faces of the crystal, — so that all our reasonings may be 

 confined to planes passing through the origin of the co-ordi- 

 nates. 



In order to represent any face, Mr Whewell encloses be- 

 tween parentheses the reciprocal co-efficients of the three co- 

 ordinates of its equations, with semicolons between them. 



