RADIATION CHARACTERISTICS OF AN ANTENNA. 221 



Treating these several integrals separately, we have 

 / "sin"-! Hi^ = I 'sin"-' H^P + / sin"! ^# 



t/o '/o *Jir 



2 



= / "sin"-' ^l^d^p + / "cos"-' xpdxl/ 



9 \ 2-4 -6 n - 2 i 



(67) 



by B. O. Peirce's Tables, Formula Xo. 483. 



Likewise * 



•^'"■'"^''* = -•!i.3.,5:7:^mf- (*«> 



Now bv Byerlv's Fouvicr\s Scries and Spherical Harmonics Equation 

 (9), Art.*122, 



n-l / 



1 - Vx 1 



Psin"-' ^P COS {B cos .A) rf^ = „_i ^ /n-l (B) (69) 



where J n-l (jB) is a Bessel's Function of the order [n — \) /2, and 



- 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 



Ji sin"-' \{/ cos xj/ sin (B cos xp) dxj/ 

 o 



B T'^ 



= - / sin"+' xP cos (B cos xP) # 



71 Jo 



n+l 



;/ n+l 



5 - 



Jn + l{B) (70) 



