- PEIECE. — BEHAVIOR OF THE CORE OF AN ELECTROMAGNET. 175 



The function H thus defined falls under theorem (V) above, and it is 

 evident that we ought to seek, for a single wire, a function ts which 

 within the wire shall satisfy (65), at the surface shall fulfil the 

 condition ^ 



and which when # = shall have the value H^ and when t is infinite, 

 the value zero. When we have to deal with a single wire of radius 

 b (= a/n) alone, it is obviously -"convenient to use polar co-ordinates 

 with origin at the point where the axis of the wire cuts the xy plane, 

 and if we do this (65) and (67) take the forms 



Cm 



Tt 



, 4 7rA^2 



w 



or cr 



p C V dm~\ , ^ 



where I, k, n, and b are given, positive constants. 



If we attempt to find a solution of (70) in the form of the product 

 of a function of t, and a function of r, we arrive, of course, at the nor- 

 mal form 



e-^'' [L ■ J,{mr) + M-K, (mr)], (73) 



but Bessel's Functions of the second kind will not be needed here, 

 and we may write, 31 = 0, 



S7 = 2 I^,n ■ e-^" ■ Jo (mr), (74) 



m 



where either m or (3 may be assumed at pleasure and the other com- 

 puted from the equation 



m^p = 4 irix(S\ (75) 



If for 7)1 in the equation (74) we use the successive roots of the trans- 

 cendental equation 



M^f>)=^^-M^f>) (76) 



