462 THE ELECTROMAGNETIC FIELD. [PT. III. CH. XI. 



which is a function only of the parameter a, and we are to put 

 a = p/h when h > p and a = h/p when h < p. We can easily show 

 that J(a) = if a < 1. For 



/ (a) = Jj (a) + J" 2 (a), where 



r 



Jj(a)= log(l -f a 2 - 2a cos 

 Jo 



/ 2 (a) = f ^log 



J 7T 



(1 + a 2 - 2a cos 

 Substituting < = TT + (/>' gives 



TTT 



J" 2 ( a ) = log (1 -f a 2 + 2a cos (f)') dfi, 



Jo 



and since the variable of integration is indifferent, we may drop 

 the accent. The integral being now between the same limits as 

 in Jj we may add the integrands, giving 



J( a ) = ["log {1 + a 4 - 2a 2 (2 cos 2 </> - 1)) d<f>. 

 Jo 



Now substituting 2< = $ we obtain 



J(a) = I r\og (1 + a 4 - 2a 2 cos </>') d$ = 

 * Jo 



Repeating the process we get 



and letting ?i increase indefinitely we obtain, if a < 1, /() = if 

 /(O) is finite. But /(0) = 0. We accordingly obtain from (14) 

 and (15) the result that the geometric mean distance from a 

 circular line is, for an outside point, its distance from the center, 

 and for an inside point, the radius of the circle. By means of 

 this result we may find the mean distance from the area of a 

 ring of finite width, of internal radius R^ and external R z . For 

 a point outside the ring the mean distance is its distance from 

 the center. For a point in the space within the ring, by ( 1 1 ) 

 or (12), 



/&2 

 logp.pdp 

 -Ri 



= i TT [R? (log Rf - 1) - R? (log Rf - 1)}, 



log f = Ti f^ 2 i<>g R* - w log A - * (R? - m- 





