17»i PROCEEDINGS OF THE AMERICAN ICADEMY. 



the series will satisfy (70) and (72), and it' the coefficients can be 

 chosen as to make 



J?L m -J (mr) = H l (77) 



equation (74) will give the function sought. 



Although the development (77) is not one of those for which the 

 coefficients can he found by the usual devices, it is easy to solve the 

 problem, for such cases as are of practical interest, to any desirable 

 approximation. 



We shall find it instructive, however, to inquire first what the solu- 

 tion would be if the second term of (72) were lacking, for, in view of the 

 fact that the permeability of the iron is relatively large compared with 

 that of the air, it seems likely that in some instances, where the seric- 

 i- very convergent, this modified problem and the real one will have 

 nearly equal numerical answer-. 



We have, then, so to choose A, , fi, and m, subject to (7")) that the 

 value of the series (77) shall be // when t = <>, for all values of r up 

 to b ; and that at every instant 



— 



s + ^ v - ('-) =0. 



It is necessary, therefore, that m shall be a root of the transcenden- 

 tal equation 



J. (mb) = 2 ** '"~ n ■ mb ■ J x (mb), (79) 



w 



which may be written in other forms by virtue of the relations 



dj 



dx 



= -Ji (#), / x ■ J Q (■>•') dx = x ■ Ji(x). 



(80) 



It will be convenient to illustrate the effect of making b Bmall (and 

 therefore // large) while a is kept constant, by a numerical example 

 1 -ume that the cross-section of the solenoid is a square of I" 



centimeters side length, so that '/ = 5 : let the solenoid have 1<» turns 

 of insulated wire per centimeter of its length, and let the resistance ol 

 these l" turns be -iV.th of an ohm. so that In absolute units w = l<> 9 /16. 

 If, then, we take the specific resistance of the core to be (10 8 /327r) 



