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XLV. Note on Proofs of Elementary Theorems of Oblique 

 Refraction. By A. Evekett *. 



IN text-books on (reometrical Optics, the elementary 

 theorems of oblique refraction (i. e. where the ray is not 

 in the plane of the paper), at a single surface, appear 

 invariably to be proved by plane projectional methods. In 

 discussions of the question of deviation by prisms, however, 

 the method of projection on the sphere is generally adopted. 

 It might therefore be well if alternative proofs by this 

 method of the three elementary theorems were given in the 

 text-books, in order to smooth the way. Apart from this 

 consideration, many people find it easier to form a conception 

 of direction in space from the spherical diagram, and the 

 proofs are extremely easy to remember, since they consist in 

 hardly more than writing down the two commonest formulae 

 for solution of a spherical triangle. Since these proofs are 

 not given in the books, perhaps an illustration may not be 

 out of place. 



Suppose a ray of light to be incident at a point Q of a 

 refracting surface. From the centre of a sphere of unit 

 radius draw radii as follows : — 



(i.) parallel to incident ray, and meeting sphere at I. 

 (ii.) parallel to refracted ray, and meeting sphere at R. 

 (iii.) parallel to normal at Q, and meeting sphere at N. 

 (iv.) parallel to any given axis of reference, and meeting- 

 sphere at P. 



Fig. 1. 



P 



Since the incident ray, refracted ray, and normal are 

 coplanar, the points N, R, I lie on a great circle. Join also 

 * Communicated by the Author. 



