OF ARTS AND SCIENCES : APRIL 14, 1868. 479 



if Wx ^2 Wg are all functions of H, making the latter the independent 

 variable and dividing by dJV^, we have 



<^?',n COS l,„ difn Siu ?',„ </«,„ 



dN nni COS Jm dN n^ cos r^ rfiV (3) 



But differentiating (2) 



dim = dr,a _ 1 



and calling 



o 



COS ija sin ?•„, rf??m 7 



rija COS T-m rijn COS 1-^. dN 



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



drm , 



dN "' 



and (3) becomes 



^m = «ra ^ra - 1 — *m (4) 



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

 For a single surface 



m = 1, 4 := 0, 

 and hence 



A = a, — ^1 = tang r, 



which equals unity when tang r^ = n^ 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 



Z2 = 0!2 ^1 h = («2 ^1 + ^2) ; 



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

 tain 



sin a 



h = 



cos rj cos r2 



a being the angle of the prism. For minimum deviation 



a .2 



Ti =: - ?'2 = «i and 4 = - tang i ; 



