Doubly Refracting Plates and Elliptic Analyzers 3 



X=O k COS(tit 

 J\=£ k COS(o)t-cf> k ) 



where O k and E k are the amplitudes of the ordinary and extra- 

 ordinary vibrations respectively, and <f> k , the phase lag of the ex- 

 traordinary vibration on the ordinary. Referred to the axes of the 

 (£-|-i)th plate, the equations take the form: 



x' k+1 =O k+1 cos(wt—t) 



y* + i=- £, * + i cos ( w/ — x) 



where O k+v E k+V \p and x are functions of O v E v <f> k and the angle 

 (A,k-\-i). Evaluating them we find: 



x f ]t+1 =O k cos<atcos(k,k-{-i)-\-E k cos[ mi — 4> k ) sin(£,£+i) 



= [C> t cos(^ > ^+i)H--£ , A cos^) J! .sin(^,^+i)] cosW 



+ iT jt sin<£ i sin(/£,/£+i)shW 

 = O k + , cos if/ cos w/+ O k + , sin \p sin <at 



y\ + = — O k cos<*tsm(k,kA-i)-{-E k cos((>>t—<t> k )cos(k,k+i) 



= [— <9 t sin(/£,£+ 1 ) + E k cos cj> k cos(£,£+ 1 )] cos wt 



-\- E k sin(f> k cos(A,k -\-i) sinW 

 —E k+1 cosx cos oit-\-E k+l sin x sin W. 



This gives the values of O k+l cosip and O k+1 s'm\f/, and of i^ +1 cosx 

 and ii ft+1 sinx. Squaring each and adding corresponding pairs 

 eliminates the phase angles and gives the values of 0' 2 k+1 and E' 2 k+V 



0\ + =0\cos^k,k+i) + E 2 k sm i {k,k+i)+O k E k cos<t> k sm2(k,k+i) 

 E\ + =0 2 k sin^,k+i) + E\cos\k,k+i)—O k E k cosci> k sin2(k,k+i). 



In passing through the plate (k-\-i) both ordinary and extra- 

 ordinary vibrations are retarded. The emergent light will be 

 given by : 



X k + l=O k + l COs(u>t— ^— 27rN 0k + l ) 



v k+l =E k+l cos(u>t~x— 2irN £k + i) 



where N 0k+1 and N £k+l are the orders of the plate for ordinary 

 and extraordinary waves measured in wave-lengths. 



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