OF ARTS AND SCIENCES : APRIL U, 1868. 479 



if «! ?i 2 n 3 are all functions of iV, making the latter the independent 

 variable and dividing by dN, we have 



di- m _ cos i m di m sin r m dn m 



dN ~~ n m cos r m dN n m cos r m dN (3) 



But differentiating (2) 



and calling 



di m = dr m _ i 



sin r m dn 



■ m ""m 7 



« m? — -r, T = 



n m cos r m m ' n m cos r m dN 



also the dispersion of a ray after passing m surfaces, or 



dr m , 



dN ~ m ' 

 and (3) becomes 



l m = a m l m _ 1 — b m (4) 



This formula by successive substitutions may be applied to any case. 

 For a single surface 



m = 1, l Q = 0, 

 •and hence 



^i = a i — \ = tang r^ 



which equals unity when tang r x = « x or at the angle of total polariza- 

 tion. That is, the unit of dispersion is that produced by a single sur- 

 face when the ray is in the position of total polarization. For two 

 surfaces (4) becomes 



k = «2 k — h = — («2 h + h) 5 



in a prism n 2 = - making suitable substitutions, and reducing we ob- 

 tain 



, sin a 



h — — 



COS T\ cos r-2 



a being the angle of the prism. For minimum deviation 



r 1 = - r 2 = ^ 1 and < 2 = - tang % ; 



