H. S. U liter — Deviation Produced by Prisms. 393 



therefore, since 7,>i7r and 7 2 '>-j7r, 

 cos E=v 2 cos p sin (X + ifi) sin (A— J/3) 



+ v sin /3 sin (X + */3)Vl - v sin 2 (A — \P) 

 - v sin /3 sin (A - J/3)\/l — v a sin a (A + ifi) 



y o) 



+ cos/3A/[l-v 2 sin 2 (A + ^)]Ll - v 2 sin 2 (k-tfi)] J 

 From (3) and (4) 



sin 3 c cos Z> = sin 2 77/ + (sin 2 c — sin 2 77/) cos i?, 

 hence 



sin 2 c cos 2> = sin 2 77/ + cos (3 sin (A + 1 /3) sin (A — £/J) cos 9 *// 



+ sin sin (A + i/3) cos 17/Vcos 9 (A — J£) cos 2 77/ — cos 2 c 



— sin ft sin (A — |/3) cos ti/Vcos 8 (A + ^P) cos 2 77^ — cos 2 c 



+ cos /3a/[cos 2 (A — i/3) cos 2 77/ — cos 2 c][cos 2 (A + %P) cos 2 77/ — cos a c] j 



1 



This is the equation of the A77/Z) surface, in the sense that all 

 values of D which correspond to physical reality must satisfy 

 relation (8). Inspection of (7) and (8) shows that, when 77/ is 

 kept constant, E and D are symmetrical with respect to the 

 locus A = 0, i. e., the great-circle H/LH of fig. 1. [This prop- 

 erty may also be deduced by simple geometrical considerations.] 

 This symmetry shows that E &nd JD have stationary values for 

 A = 0. That these critical values are minima is an immediate 

 consequence of the following premises : (a) the form of (7) 

 does not alter when v assumes different values, such as v = n 

 and v > ?i y (b) a prism of index n and angle /3 with a ray of 

 obliquity 77/ is equivalent to a prism of greater effective index 

 v and angle j3 with a ray of zero altitude ; (c) when 



* = 0,(7/ 



—7, 



— 1 



|/8), the ray in a principal plane exper- 



iences minimum deviation, as is very well known. In fig. 1, 

 Q P represents the minimum of all deviations along the small- 

 circle UY, whose altitude, LM, has any constant value. The 

 greatest deviations corresponding to this altitude and having 

 physical meaning pertain to the points U and V. 



To find out how the minima of D vary with 77/, (along the 

 arc H'LH), it is only necessary to put A = in equation (8) 

 and to differentiate D Q with respect to 77/, where D Q denotes 

 the value of D when A = 0. Then 



1 

 sin^ 



dD n _ 2 sin /3sin+/3(cos \p cos 77/ — Vcos'-^jScosS?/ — cos 2 c ) : 



drji 



sin 2 c sin Z> Q/ y/cos 2 \p cos 2 77/— cos' 



