1896.] deviation of a ray of light by a prism. 197 



Supposing these conditions to hold, we must find whether the 



value of $ given by 



1 + cos 2 .4 



cos 9 = cos 7 . —= -r— 



r ' 2 cos A 



corresponds to a minimum or a maximum value of the maximum 

 deviations. Denoting the expression 



a sin 6 n 



cos 7 cos A — — , + cos V 

 sin 9 



by M*, we have 



fl<$r sin 



T7 = 77i 7T— = T l"cos 7 cos A sin <f> — sin sin 7I. 



d<p cos (0 + 9) sin 7 cos 7 cos A L r ' 



The quantity in brackets is zero when cos <f> = cos 7 . — ^ -7— , 



positive when <£ is less than the value given by this equation, and 

 negative for greater values. The value of </> given by the above 

 equation corresponds then to a maximum value of 'SP and con- 

 sequently to a minimum value of the maximum deviations. The 

 absolute maximum deviation corresponds therefore to that one of 

 the points C and A for which the deviation is greatest. 



7T d& 



If A > 7c , -JT is negative for all values of <£ and therefore M^ 



diminishes from to A. The absolute maximum deviation cor- 

 responds therefore to the point A and is given by 



sin-2=V/A a -l sin-g. 



If J. > tan -1 (2 tan 7) the quantity in brackets is always negative 

 and the absolute maximum deviation again corresponds to A with 

 the same value as for the last case. 



As examples of the first case, which is of greater interest than 

 the others, take two glass prisms one of 20° and the other of 50°, 

 and assume the critical angle to be approximately 42°. With 

 this value of the critical angle the angle of the prism must be 

 < 60° 57' for a stationary value of the maximum deviations to 

 exist. For the first prism the deviation corresponding to the 

 point A is found to be 22° 24' and for the point G 35° 50'. The 

 maximum deviation is accordingly 35° 50'. For the second prism, 

 on the other hand, the deviation corresponding to A is greater 

 than that corresponding to C, the values being respectively 56° 23' 

 and 46° 3'. It is easy to write down an inequality involving the 

 angle of the prism and yu, separating cases where the absolute 

 maximum deviation corresponds to the point A from those in 

 which it corresponds to G, but it does not seem capable of concise 

 expression. 



VOL. IX. PART III. 15 



