176 PROCEEDINGS OF THE AMERICAN ACADEMY. 



the series will satisfy (70) and (72), and if the coefficients can be so 

 chosen as to make 



00 



^L,„-J,{mr)=H, (77) 







equation (74) will give the function sought. 



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

 coefficients can be 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 series 

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

 nearly equal numerical answers. 



We have, then, so to choose Z,„, ^S, and m, subject to (75) that the 

 value of the series (77) shall be Hq when ^ = 0, for all values of r up 

 to h ; and that at every instant 



o -\-r^ 



2 77?? W 



^'(l),. = «- (-) 



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

 tal equation 



J^ (mb) = ^ — - -mb- Ji {mb), (79) 



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



dx 



= -Jiix), I a- ■ Ja (x) d,r = a- • Ji(.v). (80) 



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

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

 Let us assume that the cross-section of the solenoid is a square of 10 

 centimeters side-length, so that a = 5 ; let the solenoid have 10 turns 

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

 these 10 turns be t^j-th of an ohm, so that in absolute units iv = 1(>V16. 

 If, then, we take the specific resistance of the core to be (10V327r) 



