Polarizing Prisms. 357 



The first term 



But 



1 C n f sin 2 <fr sin 2 </> 1 , 



" 2 J il-(a + 6cos<^) + l+(a + 6cos0)J 9 * 



f * sin 2 </># == ( ^(1- cos 2 </>)<ty 

 J c + dcos<£ J c + dcos$ 



- r^i c — cos$_ c 2 — c? 2 i 



JotS 2 " ~T~ d 2 (c + dcoscj>)i * 



'-d 2 



ire cr—ar z ir 



~~d 2 d 2 <y^d?r 



if c is > d, 



Hence 



I 



l-a- & to S j,^ = ^ 1 -"-v / (l-^ + « 2 -^)}. W 



for we can easily show that c is > d in this case. And 



r 



And the required integral is 



^{2-v/(l~2a + a 2 -6 2 )— v/(l + 2a + a 2 -6 2 )-6 2 }. . (6) 



But 



a?—b 2 = cos 2 acos 2 /3— sin 2 a sin 2 /3 = cos 2 a + cos 2 /3— 1. 



Hence, since the positive sign is to be attached to the roots, we 

 have, if /3 be < «, 



Intensity required 



7rp 2 sin /3,_, _ v , _ . ..,„.,> 



= . 2 — {2 — (cos p— cosa) — (cosp + cosa)— Sln^/3sln z aj■ 

 =7^p 2 sin^(l-cos/3){ sI ^-(l+ cos/3)] 



= 4-7rp 2 sin/3 sin 2 |-| cosec 2 a— cos 2 ^ J- (7) 



