REFLECTED AT THE SURFACE OF A CRYSTAL. 57 



Lastly, let us suppose the incident light to be polarized ; then is no longer 

 indeterminate, and equation (12) gives 



- 

 cos(p < 



which coincides with the value of the inclination of the plane of polarization to 

 that of incidence given by FRESNEL'S Theory. (See AIKY'S Tracts, p. 301.) 



SECTION III. MR M'CULLAGH'S HYPOTHESIS RELATIVE TO THE NATURE OF 

 CRYSTALLINE REFLECTION ON COMMON LIGHT. ' 



In this section we propose to determine the polarizing angle, and the planes 

 of polarization, by means of the hypothesis that each ray within the crystal is 

 produced by a portion of the incident light polarized in a certain plane. Let us 

 take the extraordinary wave, and, in discussing it, let us suppose no other to 

 exist. Let I, R, I', R/ be the incident and reflected vibrations in, and perpen- 

 dicular to, the plane of incidence. T the transmitted vibration in a plane which, 

 by FRESNEL'S Theory, passes through the axis of the crystal. Let this plane 

 make the angle 6 with that of incidence. Then all our equations of motion ap- 

 ply, if we omit T'. 



By equation (IV.) we get p^l. 



Also, we know that F= M tan <p. 



The only quantity which we cannot determine is F, ; call it M s tan $. 



Then we have the following equations : 



(I-R) sin $ + R,= T sin 0,0030 + 1, .>..>;.(!) 

 (I -f R) cos = T cos (/>, cos 6 ....... (2; 



F-R'=Tsin0 ..'.... ", ....... (3) 



R,+ T,=0 . . ...... . (4) 



o j^ i 



T,*tan0 . (5) 



I' + R'=Ttan0cot0,sin0 ...... (6). 



Adding (1) sin $ to (5) cos <, we get 



I-R = T^-cos0-T,(*-l)sin0 ($) 



sin cp, 



T , D T- cos <t>, a 

 I + K = T 2_ cos 6 



cosip 



2 1'= T (tan (/> cot $, + 1) sin 6 

 2 R' = T (tan </> cot </>,-!) sin 6 



sinc/> 

 - 



\ cos 

 VOL. XV. PART I. 



$, sinc/>\ ,. 



' : 7- 1 cos a+T, (s l)sm <p. 



(j) sin 07 



