110 Professor Airy on the 



The coefficient of cos £-(a+0) is 



(1 — k) . cos d> + k (1 + k) . cos cj) . cos — (1 + k) . sin <p . sin — - — . 



X X 



Tlie sum of the squares is 



1 + k°J -Zk.l-k 2 . cos 20 + 2£ . 1 - &*. cos 20 .cos — — 



?4 . . 2tt6 



— (1 - A) . sm 2d> . sin — — . 



A 



Restoring the factor , ,„., , we have for the brightness 



2) 1 ~ ( iUt cos 2< ^ + a+*r cos "♦•"■"x 



1-A' . ^ • 27r9l 



1 +k~ 

 If we make tan x = „, • tan 20, the two last terms, or 



i-A 2 c 2* 2*-e . . 2*-e? 



i+FiTTF cos ^- cos -x- ~ sin 20-sm— j 



l-/r ./ 4A 2 , . . _ fflJL 2x9 



become /2 V , t ,., v cos- 20 + sin- 20 . cos — — + x> 



1 +re (1 +A"j- A 



and the expression for the brightness is 



J 1 - IT? (^aw- 2 ^^ 2 * + r+V 082 *) 



+ 2i^,\/j^~^h^ . cos^ + || . 



6 



1 st . As the multiplier of cos 2 — + ^ is never = while /• has any 



X A 



value between and 1, the rings are not interrupted in any part of their 

 circumference. 



