PEIRCE. — 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 zj which 

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

 condition 



■* + — (A ~ "' B) if + -^r-JJ s • ** - * (69) 



and which when t — 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 acy plane, 

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



Tt 



i^7w,L' ■>;_!• (70) 



., + ^-^*! + »^ ^ =0, (71) 

 w at w \drjr=b 



where /, 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 



erf™ [L ■ J (mr) + MK I} (mrfl, (73) 



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

 and we may write, M = 0, 



ST 



= >L, n -e-^ l -J (mr), (74) 



where either m or f3 may be assumed at pleasure and the other com- 

 puted from the equation 



m 2 p = 4 ttixP 2 . (75) 



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

 cendental equation 



Jo (mb) = i __ ■ J, (mb) (76) 



