576 PROCEEDINGS OF THE AMERICAN ACADEMY. 



w'G-\- (.4 -n^B) ^irN"" ■ ^ + t^^^ff^ ■ ^'^ = E, (42) 



and if Hg represents the strength of the magnetic field in the air space 

 within the solenoid, and A — irB is written h- A, 



where the double integral is to be taken over the cross-section of a 

 single filament. If we put H = H' -\- H^ and Hs = H's + H^, the 

 last equation becomes 



H's + ; -rr + —^^ / / -^■'^^^ (44) 



w at w J J dt 



in which //' satisfies at every point the equation 



~ 47r/x V dx' "^ df J 4/.7r ( r'dr \ ' dr ) \ ^ ^ 



dt 



where r is the distance from the axis of the wire in which the point 

 lies. We are to find a function H' which satisfies ecjuations (44), (45), 

 which, when ^ = 0, is everywhere equal to Hg — H^, and which van- 

 ishes everywhere when t is infinite. 



If C7 = 2 L^^'' ■ -^0 {mr), (46) 



in which either m or ^ may be chosen at pleasure and the other com- 

 puted from the equation 



m^p — 4 7r/x;8^, (47) 



and if for m in (46) w^e use the successive roots of the transcendental 

 equation 



Jo (mb) ( 1 ^ J ^ ^^^ • mb ■ Jx (mb), (4S) 



where b is the radius of the wire, ct satisfies ecjuations (44), (45) and 

 vanishes when t is infinite. 



Without any consideration of the question of a possible development 

 of unity in terms of an infinite series of Bessel's Functions of the form 

 J^) (jnr). where the w'shave the values just mentioned, it is clear ^ that, 



* Byerly, Annals of Mathematics for April, 1911. 



