376 APPENDIX CALCULATION OF THE 



whence - = 4 = tan (ISO /') = 21 48' nearly, 

 5 



hence /' = 180 (2 i 48') = 158 12'. 



By this method we may in all similar cases deter- 

 mine any required angle, whatever may be the ratio 

 ofag: af. 



Where the planes a c, a b, are not at right angles 

 to each other, the angle c a 6, may be either acute 

 or obtuse. In either case knowing the angle c a 6, 

 and the particular values of a c and a b, deduced 

 from the known ratio of m : n, and from the given 

 law of decrement, we may obtain the angle a b c from 

 the formula 



i , d'. tanff. \ s 

 tang. \d -f~ 



where d =z difference of required angles, 

 d' z= difference of given sides, 

 s zz sum of required angles zz: 180 given 



angle. 

 s 1 zz: sum of given sides. 



Where spherical triangles have been used for de- 

 termining the law of decrement, they may also be 

 used for determining the angles of the secondary 

 planes with the primary, the law of decrement being 

 known ; with this difference however, that where in 

 the former examples we have sought the sides of 

 those triangles, knowing the angles, we have now to 

 determine the angles from the given sides : and the 

 sides are known from the plane angles of the primary 

 crystal, and from the ratio of the edges of the defect 

 of the primary form, as deduced from the ratios of 

 the corresponding primary edges, and the law of de- 

 crement. 



