Surface-Film in Total Reflexion. 15 



each o£ the directions Hj and H 2 ; i. e. we must write 



E ( {1 — /r} sin T 4- k 2 sin T) + E'(/ccos T' — /ccosT') in the direction Hj, 

 and 



E (/ecosT — /gcos T) + E'((1 — k?) sin T' + * s sinT') in the direction H 2 . 



k is very small. The underlined terms are right-handed or 

 negative ellipses. Let the right-handed vibrations receive 

 the phase-retardation i\ in passing through the first plate, 

 and the left-handed vibrations the phase-retardation l { . Then, 

 if we disregard the k 9 terms, after the first plate the two 

 components are 



E sin (T-v%) + W(k cos {T -l x ) -k cos (T'-^)) 



and 



E(«cos (T-vO-k cos (T-/ 1 )) + E / sin (T'-l,). 



These may be written 

 (E^ + E^^sin^T-n-tan- 1 -^^ + E'*co3 (T'-J,), (I.) 



and 



(E^ + E^isii^r-^-tan-^WE/.cosCT-.'O. (II.) 



Let r 2 and l 2 be the corresponding phase-retardations pro- 

 duced by the second plate. Then, if we consider only that 

 portion of the phase-retardation which varies with the thick- 

 ness of the plate, we shall have, after the second plate, in each 

 of the directions H^ and H 2 , four vibrations with the phase- 

 retardations 



r x +r 2 , '>\+I. 2 , h + r 2 , and l x + l 2 . 



Hence we shall have vibrations with those phase-retardations 

 in the principal plane of the analysing nicol. In the ex- 

 pression for the resultant intensity there will occur a number 

 of terms, of the form 



Acos{(r 1 + r 2 )-(/ 1 + / 2 )+6x}+Bcos«!(^i + ? , 2)-(> J i + y +l>\ 

 + Cco^{r 1 + r 2 )~{l l + r 2 ) + c\+T)cos{{r 1 + l 2 )-{l l +r 2 )+d} 

 + E cos \ (r, 4- h) - {k + k) + e\ + F cos { {l ± + r 2 ) - (7 X + h) +/}. 



Write r l — li=B 1 and i\ 2 — l 2 =—8 2 . Then we may write 

 the six phases 



$,-& + «, -S 2 + & ; & + c, Si + g 2 + ^, & + «, and -&+/. 



