430 Prof. Rowland on the Propagation of 



But 



d __dp d TLr sin cos a d 



doc doc dp p dp 



Whence 



d/2^ + 0, \ _ Rr g . n Q cog a 3 cej 



doc\ p J p L 



Whence we have 



N' = ^ 1 — V~ COS 2 Wa. 



*>*" Jo p 2 



In the expansion put 



v" 

 p = R; g= — cr sin sina + c^-n- 



^ = £ { C l( R) + f C,(E) + ^ C,(R) + &c. } . 



Writing 9 



# == 9uj an( i ' l== ~~ cr si 11 0> 



2R 

 we have 



q=g + h sin a, 



T 27r f 1 «fw — D 1 



j s » cos 2 ocdoc = 2 | ^ n + Y 2 n 9 ^ 2 ^ 



+ 1.2.3.4 2.4.6^ /i+to - 



/»— 1).. . (n-2m + l)1.3...(2m-l) „ . 79 1 

 1.2. ..2m 2.4...(2m + 2)^ + <^.j>. 





n 

 + 



So that the problem is completely solved without any approxi- 

 mation and for all distances at which the series is convergent. 

 At a great distance the expression becomes very simple. 



3cWC smf f 1 J_ 



iN 4ir ]i 1 2 + 2 2 . 4 S n " 



+ (2rb r 6 crVsin ^ +&c ' 



+ ( 2.4...2n)'(2» + 2) ^ sin " ' + &C ' } 



We recognize this series as a BessePs function divided by 

 hr sin 0, and so we can write 





