210 A. Mukho-pudhjaj—MUptic Functions and Mean Values. [No. 2, 



Since the angle FOR = i)/, we have to find the average value of 2i|/. 

 If, therefore, P be the average value required, we have 



dr. 



Jh 



whence 



Jf^a 



P -,r—a y^r — a 

 l{a-l)V= r + - I rd^. 



^ -^r=h Jr = b 



i (a- 5) r= I ^ dr. 



Integrating by parts, we have 



But, from the formulas in § 1, we have 



tan, = ^ = ^(«^T, 

 ^ a a \r^-h^J ' 



SO that, when 

 Therefore, 



Assume 

 so that 



and, accordingly, 



r = a, tj/ = 0 



J^r = a 

 rd<p. 

 r = h 



r2 = cos3 1? + 62 sin^ 17 



aS - r2 = (a2 - £2) sin^ -q, 

 5-2 - &2 = (a2 - 62) cos2 77, 



tan i(/ = - tan n 

 a 



sec2 )(/ cZ i|/ = - sec2 -q.dr] 



a2 cos2 1; + 62 sin2 )7 



rcZ^ = ^ (Zjy. 



(a2cos2 57 + 62 sin2 7;) ^ 



