RADIATIOX CHARACTERISTICS OF AN ANTENNA. 221 



Treating these several integrals separately, we have 

 / "sin"-i H^ = I "'sin"-' iA# + / sin"-i i^d^ 



2 



T IT 



= / "sin"-' \PdiP + / "cos"-' ipdi 



"^ ) 1-3-5 n - 1 f 



by B. O. Peirce's Tables, Formula No. 483. 

 Likewise 



(67) 



JfTT 1 9. A. a. . • -n I 



' sin"-i H^ = 2 ; J^ ^-— . (68) 



/ 1 • 3 • 5 • • • • 7i + 1 ) 



Now by Byerly's Foiirirr'is- Scri<'.s and Spherical Harmonics Equation 

 (9), Art.'l22, 



2"^v.rf? 



I sin"-' rP cos {B cos ^) d4^ = „_i ^ ' J n-i (B) (69) 



B 2 2 



whereJn-i (B) is a Bessel's Function of the order {n — I) /2, and 

 r f - 1 is the Gamma Function of - • 



For the last integral of (66), we have by Problem 2 and Equation 

 (9) of the same article of Byerly's Fourier's Series 



J sin"-' 4/ cos i/' sin {B cos i/') d\p 

 o 



B T'^ 



= - / sin"+' ^p cos {B cos i//) # 

 n Jo 



n+l / 



n "+' 



./.+i(J5) (70) 



fi 



•> 



