512 PROCEEDINGS OF THE AMERICAN ACADEMY. 



of the length of the core, when the core is without currents and there 

 are no other circuits in the neighborhood so that the magnetic field 

 within the coil is uniform, this inductance will be increased by an amount 

 U when, in consequence of Foucault currents running right-handedly 

 around the axis, the intensity of the field within the core is raised above 

 4 7r N C The contribution L' comes from a field every line of which 

 threads every turn of the coil. We have, therefore, 



U = Nf]i (ff- H a ) 2 irr dr = 2 Tr ft N f'lfr dr - 4tt* a* N 2 n C, (7) 



and if the coefficient of C in the last term be denoted by L u the whole 

 induction flux through the turns of the coil per ceutimeter of the length 

 of the solenoid is 



p=(L- Zj) C-\-27TfxNfHr.dr; (8) 



Jo 



if to is the uniform resistance of the coil per centimeter of the length 

 of the core and E the impressed electromotive force in the coil circuit, 

 reckoned in the same way, 



*-£ = .<* m 



dP P a 1 TT 



or E=wO+(L-L 1 )-^+2^ixNj °L..r.dr; (10) 



or, by virtue of (6), 



E = wC+(L-Ld*£+±N P a (|f )_• (U) 



To fix one's ideas, one might imagine every centimeter of the length of 

 the coil (measured parallel to the core axis) to be a separate circuit con- 

 taining an applied electromotive force of E absolute units (the same for 

 every such circuit) and having a total resistance w made up of the resist- 

 ance (w') of the wire actually wound on the core, and the resistance (w' ! ) 

 of the external part of the circuit which is non-inductive. 



If, in order to determine a set of normal special solutions of the linear 

 equation (6), we assume H to be the product of a function (T) of t 

 alone, and a function (R) of r alone, and substitute this product in the 

 equation, we arrive at the well-known normal form 



- a , t 



[A -J (nr) + B- K (nr)l (12) 



