L. B. Tuckerman 



COS2(o,l) COS2(l,2) 



l ~\~ e \ — sin2(o,i)sin2(i,2)cos27riV^ 



2 € 



-\-P .".j, sin 2(1, 2) sin2 7rA r 1 



i-j-e 



P 



i-f-<? 2 

 2e 



+ cos 2(0,1) cos 2(1,2) 



— sin 2(0, 1 ) sin 2 ( 1 , 2 ) cos 2 irN\ 



' — — V sin2(i,2) sin27rA r '. 



Dividing through by P g .," cancelling like terms and putting 



2e 



:tg2 



sin 2 (1,2) 



=0 



1 — e\ 



sin 2(0, l)(COS2 7rA r , — COS2 7rA r ' 1 ) 



— /'(sin2 7rA r 1 — sin2 7rA 7 ' 1 ) 



If sin2(i,2)=o, it is a match, for all values of t, N v N\ and 2^(0, 1), 

 hence the system is useless. If sin 2(1, 2)^0: 



sin2(o, 1) (cos27rA r 1 — cos2ttA 7/ 1 )- — /(sin27rA^ — sin27r JV\)=o 



Writing N=j4(N l + A 7 ^), \N=y 2 (N l —N' l ), expanding, collect- 

 ing terms and dividing by — cos2 7rA r sin2 7rA N: 



^H~sin2 (o, i).tg27r A/=o (52) 



It is evident that, with a known average order, N , of the half- 

 shade, a match enables one to determine the ellipticity of the 

 incident light provided the azimuth of its major axis be known, 

 and conversely to determine the azimuth of its major axis pro- 

 vided its ellipticity be known. 



In measuring ellipticities such a system will give a match only 

 when /<tg2 7riV, but since in measuring ellipticities a compen- 

 sator is always used to increase the sensibility, this limitation is 

 of no significance. 



Balanced Half shades. — A balanced half shade is defined as one 

 in which N=o and therefore tg2 7rA r =o. Such a half shade is 

 especially convenient from a theoretical standpoint, since it gives 



184 



