374 APPENDIX CALCULATION OF THE 



The preceding sketch of the methods of discovering 

 the laws of decrement, will, it is hoped, be generally 

 found sufficient for that purpose, wK" the angles at 

 which the secondary planes incline on the primary 

 are known, and where the ratios of the edges or 

 other lines already described are also known. 



But it very frequently happens that the whole of 

 the primary planes are obliterated by such an ex- 

 tension of the secondary planes, as.produces an entire 

 secondary crystal. In these cases we must recur to 

 cleavage for determining the relative positions of the 

 primal^ and secondary planes, and for measuring 

 the angle at which they meet. The cleavage planes 

 which we may adopt as the primary set, if more than 

 one set be discoverable, should be those which are 

 most compatible with the observed secondary forms. 



Having thus given an outline of the solution of our 

 first problem, by shewing how the laws of decrement 

 may be determined from certain data, we shall pro- 

 ceed to examine the second, and to ascertain how the 

 angles may be determined at which the secondary 

 planes incline on the primary, the elements of the pri- 

 mary form., and the law of decrement, being known. 



The methods used for determining these angles, 

 will be nearly similar to those already described for 

 determining the laws of decrement. 



The plane triangles which have been used for 

 determining- the laws of decrement, have been both 

 right-angled and oblique. 



Where a law of decrement is expressed by means 

 of the ratio of the sides of a right-angled triangle, the 

 angles are readily found by reducing the ratio to that 

 of radius, and tangent of the required angle. 



