Proofs of Elementary Theorems of Oblique Refraction. 481 



PI, PR, PN by great circular arcs. Let 0, <£/ be the angles 

 of incidence and refraction, //,, /jl the refractive indices of 

 the first and second media. 



Then arc NI = <£, arc NR = f . 



The angles cp, </>' being acute,, from the nature of the case, 

 and the refractive indices positive, the law //,sin (/> = /// sin ft 

 shows that ,NI, NR have the same sign, hence I and R lie 

 on the same side of N, within 90° of N. 



I. From the spherical triangles NPI, NPR, we have 



cos PI = cos PN . cos (j> + sin PN . sin <f> . cos N, 



cos PR = cos PN . cos <f)' + sin PN . sin </>' . cos N. 



To get rid of N, multiply the first equation by //,, and the 

 second by // , and subtract ; then since jul sin <t> = ft / sin <j>', 



a cos PI — u! cos PR , ... ,_,. 



» CQS p N -=a*cos^-/a COS0, . . (1) 



which is independent of the position of P, i. e. of the direction 

 of the axis of reference. 



By taking as axis of reference in turn each of three 

 co-ordinate axes, we get the general equations of refraction 



oL-ll'L' fM-fi'M' /.N-VN' ^ , ., 



i III It 



L, M, N, L', M', N', /, m, n being the direction cosines of 

 the incident ray, refracted ray, and normal respectively. 



(These three equations are, of course, only equivalent to 

 two, owing to the relations between the direction cosines.) 



II. If the axis of reference is perpendicular to the normal, 

 PN = 90°, andcosPN = 0. Hence, since yu,cos <f> — fx' cos <j>' 

 is not infinite, we have from (1) in this case 



yitcosPI— fi' cos PR = 0, or fi sin 7j = fjb' sin?/', (2) 

 where rj = 90°- Pi t/ = 90° -PR. 



Hence the angles made by the incident and refracted rays 

 with any plane through the normal obey the law of refraction 

 (P being the pole of this plane). 



III. From the triangle NPI, 



sin N _ sin IPN 

 sin PI " sin IN * 



Hence, putting 7= ZIPN, 7'= ZRPN. we have 

 . XT cos 11 . sin <y 



f ) SlllN = — (— : ~. 



' ! sin 9 



Phil. Mag. S. 6. Vol. 38. No. 226. Oct. 1919. 2 L 



