46 PROFESSOR KELLAND ON THE POLARIZATION OF LIGHT 



Denote 2 (0 r + ~ 8 * 2 ) (1 - e~' ^cosf 8 y) by D ; then, by collecting all the terms, 



d 2 a 

 we obtain for the value of that part of -T-J which is written dorcn in equation 



(6), and which is the part involving 8 a 



2 (</+ 0-T **) 8a = -(I-R) sin0-^ (sin 2 0-2 cos 2 0) 



* 



?* 



4^(^-^)sin0M(sin 2 0-2cos 2 0)-DR / (a.) 



Having written down the full work of the reduction of this part of the ex- 

 pression, no difficulty will be experienced in following the rest of equation (5). 

 It must be remarked, however, that for the reflected wave 



8 x r cos +/>sin0, 8yrsm <pp cos0 ; 



for the wave T, within the crystal, 



8 x'~c cos (/; +p sin 0, 8 y'= e sin +p cos <j> ; 



and for the wave T' 



8 xf = o cos (f)+j> sin (p, y'=o sin (f)-rpcos(p. 



Hence, referring to equations (4) for the value of d /3, we get 



2 d zdy d /3= 2 --- | i cos<p+psin(p( ism(p+pcos(f))( 2Isin 2 -^ +- sin/fcz) 



Ar 1 dH *[ 



+ rcos(p 4-josin0(rsin0 /cos0)( 2Rsin 2 -^- + - -r sin Ar 1 cos0 



'' 



= -2-^- | -(/ 2 ^ 2 )si 



/* V. 



p*} sin d> cos ( 2 R sin 2 -\ --- sin kr) \ cos (b 



2 e dx ) 



M /rfl </R\ -I . 

 R)- 7 (^-^) Jsm0cos 2 



by virtue of equation (7.) . . . (b.) 



os0(-2Isin 2 -^ + - 



Jml 6 Ct AJ 



2(0/-t-i - 8 **) 8 a,, can be written down at once from equation (a), from which 



T 



it differs in no respect save that 0, and 0' occupy the place of in the inci- 

 dent pencil, and that the coefficients and circular functions are indicated by a 

 different letter. An inspection of the value of 8 a, in equation (4), will therefore 

 shew that 



2 (0 / + 1 8 y! y ] 8a,= -T sin 0, cos 6 ^ (sin 0,-2 cos 2 0,) 



c 2 1 dT 



+ T' tin 0' sin & -^ (sin 2 0, 2 cos 2 0,) + - -7 sin 0, cos 6 M, (sin 2 0, 2 cos 2 0,) 



A G . (I X 



