286 On the undulatory Time of Passage of Light through a Prism. 



in which 0' is to be considered as a function of 0, depending on 

 the prismatic connexion between the initial and final directions. 

 For an ordinary prism in vacuo, having its angle = ot, and 

 its index = ju., so that 



sin* =p,sin~, (7) 



i being the angle of external incidence corresponding to the 

 minimum of deviation, the relation between 0, 0', is, 

 p. 2 sin <cr 2 = sin (/ + 0) 3 + sin (z-0') 3 



+ 2 cos to-, sin (z + 0) . sin (i— 0'), (8) 

 if the positive semiaxis of x be an emergent ray of minimum 

 deviation, and the positive semiaxis of x' the corresponding 

 incident ray prolonged, while the positive semiaxes of y, i/ 9 

 lie on the same side of the axes of x, .r', as the prism. The 

 relation (8) may be put under the approximate form, 



m 



e' = -~.a 3 , (9) 



when the angles 0, 0', are small, that is, when we consider rays 

 having nearly the minimum of deviation, m being the same 

 positive number as in my last paper, namely, 



8 sin (*+v) sin(z— -J 

 m = _ 1 _ - (10) 



sin 2 i . (cos -5- ) a 



and if, besides, we consider the originates y> y\ as small, thnt 

 is, if we suppose the light to pass near the edge of the prism, 

 and neglect terms of the fourth dimension with respect to the 

 small quantities y, 7/, 0, we shall have the undulatory time or 

 characteristic function V = the maximum or minimum, re- 

 latively to 0, of the expression, 



X= X-,/ + (y-y>) t- i{x - x -- ?L yl) p- ILjp. (11) 

 In this manner we find, with the same order of approximation, 



V = x-x +\. KJ *} - + — . ^ ■? •?' w y ; , (12) 



x— x f .4? (x — x'Y v 



a result which may also be thus expressed : 



v= \*+Jk*-*- f w+^f ---{*=£)'. m 



If we neglected the last term of this last expression, it would 

 give, by (1) and (5), the following formula for the tangent of 

 inclination of the emergent ray, 



tan 5 = ^ • = • ?.'7$!? ; (11) 



