1889.] of the Plane of Polarisation of Light, 67 



we may write 20 for sin 2Q, and cos 4w for cos 2 (^1 + ^3), where u> 

 is the value of w when m = 0. Since 



tan 2iv Q = tan 2x cos hz, 



we have, putting a equal to tan 2a, 



, 1 — a 2 cos 2 hz 

 ,cos4^ = _ _. 



1 + a* cos #3 . 



Making these substitutions the differential equation becomes 



dQ = 2mdz — hQ - — a CQS ^ ^ an ^ _ ^ 

 1 + a 2 cos 2 &z 



or, putting p — hz, 



dQ 2m n l-a 2 cos 2 /3, Q 



— = Q ~ tan 3. 



dp h l + a 2 cos 2 /3 M 



Put P == - — C \ C0S o @ tan /3 for brevity, and the integral of the 

 l + a 2 cos 2 /3 ' J ' & 



equation becomes 



. f e' Fdft dp. 

 Jo 



It is easily shown that 



W 2m 

 lie = — 



. l + a 2 cos 3 /3 



Pdtf =.log— -— 



H 8 cos 



.-. e /Pt ^ = sec/3 + a 2 cos 



Substituting we find 



o _ 2m J 



J n 



sec /3-f-a 2 cos ft) dp 



h sec /3 + <x 2 cos /3 



2m log e tan (jTr + i^) + a 3 sin hz 

 k sec &2 + a- 2 cos &z 



It appears from this equation that Q is a periodic function of z, the 

 wave-length being tt/Jc. 



If we make h very small, we may put log tan (\ir + ^Jcz) = hz, and 

 the equation becomes 



_ 2m hz + cfikz 

 ~ ~h 1 + a 2 



