IN FOCAL PLANE OF TELESCOPE. 343 



The effect due to the remaining part of the circular source can 

 be easily calculated by adopting the semi-convergent series for 

 J" e 2 (/>)+D r 1 2 (jo). Retaining the first term only, we get for the illumina- 

 tion at an internal point due to the part external to the sector. 



Tz --(2«-öVo V L-Jc? siri J <p / ^/1—Jc? sin 2 tp 



^ 7T-a 2 / E(ft) Ay sin 2ft \ 



7t 7t\2a-d)\ k/ ~ tJy ^ 1 ' 2k f */l-k? sin 2 ft / 



where F(ft) and E(ft) denote elliptic integrals of the 1st and 2nd kind 

 respectively. 



For an external point, we shall have 



r.-S- 



i E(ft) . _, .. k; sin 2ft \ 



\ fc e 2k e ^/l — Je? s in 2 ft/ 



n\2a— ô) 



Thus by the addition of these separate effects, we obtain the 

 expression for the intensity of illumination at an external point in 

 the very neighbourhood of the rim. 



- l ~ ^la-ôA k\ +F{ ™ 2k\ 



1+Ja[a—d) 

 At an external point 



2 / E(ft) „, , k? sin 1ft \ 



e e s ^-(2a — d)\ A- e vriy 2& e ^/l — k/smftj 



I's' 



4- —/ /' (VI) 



The expression within the brackets can be calculated by means 

 of Legendre's tables for the two integrals E and F. The course of the 



function for different values of </>. 2 and -~ — ™ shown in Fig. 5 PI. XVII, 



