512 Prof. J. I). Everett. [Jan. 22, 



Let X fxv denote the direction-cosines of the tangent to the refracting 

 surface in the plane of incidence. 



Then l 2 m 2 n 2 may be regarded as the projections of a unit incident 

 ray on the axes. But this unit ray gives a projection sin x on the 

 tangent, and a projection cos x on the normal. Adding the projections 

 of these on the axes, we have 



U = X sin x + h cos x 1 



mo = /x sin x + «h cos x r 



n-2 = v sin x + n\ cos x ^ 



And in the same way, by projecting a unit refracted ray, we have 



V = X sin x' + h cos x' "1 



m = fx sin x' + nil cos x' r 



ri = v sin x + n i cos \ ^ 



V, m', ri denoting the required direction-cosines of the refracted ray. 

 Substituting the values of A, v from (1), equations (2) become 



V = Jc (h - lj_ cos x) + h cos x' "1 



m = Jc (m 2 - mi cos x) + r »h cos x' f (3) 



ri = Jc (no - ni cos x) + n\ cos x' J 



Jc standing for sin x7 sm X the relative index from the second medium 

 to the first. 



Equations equivalent to (3) are given in Herman's Geometrical Optics, 

 §§ 18, 19 (Camb. Univ. Press, 1900) and have, I understand, been 

 taught for many years by Mr. Webb at Cambridge. 



15. In the original calculations for this paper, a clumsier method 

 was used, in which X /x v were computed by means of deter- 

 minants.* The original results have been tested, and in some instances 

 made more exact, by the use of equations (3). 



* Since the tangent is coplanar with the normal and the incident ray, we have 

 AA.+ Bfi -+ Cv = 0, 



rhere 



1 1 m l 



1 m \ n \ 1 



B = | 





1 m. 2 n 2 J ' 







c = 



Also, since it is perpendicular to the normal, we have l^ + m^ + n x v = 0. 



Hence A, fx, v are proportional to the three determinants 



t I -BO I ^ I C A I I AB I 



L = I, M = 7 I , IN = , i, 



and X, fx, v are the quotients of these by V(L 2 + M 2 -rN 2 ). V, m' , ri are then found 

 by substituting the numerical values of A, /x, v in equations (2). 



It can be shown, by expanding the determinants and making obvious reductions, 

 that \/(L' 2 + M 2 + N 2 ) is sin x, and that the final results reduce to equations (3). 



