518 PROCEEDINGS OF THE AMERICAN ACADEMY. 



satisfies (23) provided that n a is a root of the equation 



71 ft S 



J (n a) = — — - J x (n a), (25) 



and if we use the successive values 1:L of n 



2-J (n p r) 



so that 



and 



i = V ^»M (26) 



a ^ W/ ,(l+* a VM (»,<*)' K > 



H * = —2-,\ 2— 2 - (28) 



a *4 1 + s 2 n p 2 ' v } 



In the case here treated n a is a root of the equation 



^(*) = !fjiO), (29) 



and it is not difficult to prove by the aid of Meissel's Tables that the first 

 five roots have approximately the values given below. 



TABLE V. 

 n x a= 2.2218 

 n„a = o.l 171 

 n 3 a= 8.0624 

 w 4 a = 11.0476 

 n 6 a = 14.0666 



9H 



Since 4 -k q = — -— (30) 



9 



dJ n (nr) _ , _ _ 



and — ^ - = — »■ J x {nr), (31) 



= -^ V e ~ ap2< ^ ( w r ) /Q2^ 



y 2 7T a ^ (1 + s 2 n 2 ) J^ (n a)' ^ ; 



The function defined by (27) satisfies (23) when r = a for all values 

 of t, and equation (6) for all points within the core for all positive values 



11 Byerly : Treatise on Fourier's Series, etc., p. 220. 



