58 Mr. Mac Cullagh on the Laws of 



calling this azimuth a, and putting X for the acute angle at which the axis of the 

 crystal is inclined to its surface, so that X may be the complement of the arc AY, 

 we have 



sin 5^ = cos X sin a, tanja^cotanXcosa; (54) 



and by making these substitutions in formula (53), after having changed sin t^ 

 into cosw, that formula becomes 



•nrj=7ir— A;(sinV — sin^X)-l-^sin*wcos*XsinV (55) 



which agrees with the remark of Brewster. 



The deviation 6.^ or 0^ is found from the second of equations (38), by putting 



■ f _ -. for tane, and by substituting for cos(«i-|-t2) the value (49) or (50) which 

 sin \jp t^j 



it has at the polarising angle. The result is 



e3=0'3=-^sm2qsm(p-\-,^), (56) 



since the small arc 63 may be taken for its tangent. This result is easily trans- 

 formed into 



e3=0'3= — Ksinycos^, (57) 



where <j> denotes the arc Ai, or the angle which the incident ray makes with the 

 axis of the crystal ; and this last expression is equivalent to the following, 



e3=:0'3= — KCOs\sina(sin\coST!r-f-cosA.sin7ij- cosa), (58) 



which gives the deviation in terms of \ and a. 



As an example of the application of our formulae, we shall make some com- 

 putations relative to Iceland spar. According to M. Rudberg, the ordinary 

 index of that crystal, for a ray situated in the brightest part of the spectrum, at 

 the boundary of the orange and yellow, is 1.66; and the least extraordinary 

 index for the same ray is 1.487. Dividing unity by each of these numbers, we 

 geta = .6725, ft=.6024; whence w= 58° 56'; ;f=. 1164=6° 40'; k=.1587= 

 9° 5'. Having thus determined the constants, we can readily calculate the pola- 

 rising angle and the deviation, for any given values of \ and a. 



First, let us see how the polarising angle varies on different faces of the 

 crystal. 



1 . When \ = 90°, the face of the crystal is perpendicular to its axis, and Wj 

 is independent of o. In this case, the formula (55) gives 



