R. S. U filer — Deviation of Bays by Prisms. 225 



tions OB", OB 7// , OF" and OF'", that is, all starting from any 

 common point O. Lay off OB /!r =OB ,,, =OF / '=OF ,/, =L. 

 £B'"OB' , = i 1 , ^F"'OF # =» 4 , ^B'"OF"'=D' and ZB"OF"=D. 



Since, from the general theory of prisms, i 1 =i 49 a triangle 

 B"GF" may be constructed in a plane containing B" and F" 

 and perpendicular to OG-. Join B ;// to F //; . 



Now in the triangle B"GF", B^^OF'siniD' and, in 



the triangle B"OF", FF"=2L sin^D. Also GF'=L cos i„ 

 hence 



sin -|D = sin ^D'cos i x . 



Furthermore, when cos ^<1, equation (1) shows that the devia- 

 tion of a ray not in a principal section is less than the devia- 

 tion of its projection on such a plane, whereas the text-books 

 agree in writing " I3>D r . " 



We shall next consider what bearing formula (1) has on the 

 question of minimum deviation. The well-known, general 

 equations for refraction by prisms are 



sin £ =w sin z' ) -. ( *,=£, 



. - 1 • . 2 y and ■< . 2 . 3 



sln^ 4 =?^sm^ 3 j ( ^ 1 = ^ 4 



cos ij sin /3\=n cos i t sin /3' 2 



cos i t sin /3' 4 =n cos i 3 sin /3' 3 



*' = £'. + 0', 



where n denotes the ratio of the absolute index of refraction 

 of the material of the prism to the absolute index of the sur- 

 rounding medium. In general, we shall assume w> 1. By 

 using the equations of the above list together with the relation 



cos fH\ cos )3' 8 = cos p\ cos /3' 4 ; 



which is a necessary condition that D / shall be either a maxi- 

 mum or minimum, when i x is kept constant, we find /3' 2 = 

 $' % =\a! and hence 



cos t, sin ^(D' + a') = ( + /V /7i 2 — sin"*,) sin \d. (2) 



D' denotes the possible stationary value of D'. 

 Equation (2) is equivalent to the usual formula 



"cos i, sin $(D , a + a')=n cos i 2 sin |a'," 



since n sin * 2 =sin *, and ^ 2 >-Jtt. It should be observed that, in 

 obtaining equation (2), use has to be made of the relation 

 D'=fi\ + ft\ — a' and this shows that the various writers intend 

 to employ the same definition of deviation as the one given 

 above. 



d 2 D' 



If we actually test , a , under the specified conditions, we 



