IX FOCAL PLANE OF TELESCOPE. 



335 



The latter integral should again be divided into two parts, of 

 which the one is 



dr 



whence we obtain by expansion, 



p [l -fo,-) + /r(,))l -^=^+1^+^ 



'o L \ /J ^/-ia J — r-- a a 9 a 5 



which converges very rapidly when a is large compared to .r,. 

 In the second integral, we have to put 



r2 / \ , T "f ^ 2 {^ cos%- l—sin2r\ 



JVW+JMr)-— (1 ^ +— tss— ) 



Thus, we shall have to calculate the following integrals : 



/" 2a dr L j , &! , Q x, 



/ „ = r— /or/ to —f- where ä 1 = a-rc sin — — ; 



J Xi r^/4,a?—r À 2a l/ J 2 ' 2a' 



-, / - a co,9 2/- oY -, / '-'" sw 2r dr r , . , 



and / --r-- — — - and / „ of which the last two 



y, ?-v4a 2 — ?- ./. rv-irr — r 



require a little consideration. 



/ 2a cos 2r oY _ 1 / -'* cos 2r d(r) 

 */n r 1 , v /4Â- — ?•- 2 ./ r r' ^/4a J — r* 



1 am - cos 2^ /• /" r r J /cos2r\ 



=TT\/4a-— ay ^- + a /l — — tf( — » — 



2 v a?i 3 ./ V 4a- \ r 3 / 



Expanding /l— ZL in powers of -^-j, and neglecting terms beyond 

 — 5-, we can find without difficulty 



f- a cosZrdr 0.01367 3 cos 4a 



/ ., /A ., v — tïïï v — nearly 



/ »-*/4a- — r- a 32 a- 



i a ^/4ta^—r A 



In a similar maimer 



2a srâ 2r d 



/■-" smwar u.uurf'^ 5 cos 4a . 



/ s /^ -^ ^ ~ « Î7T 3 — nearly. 



