and Inductance of a Concentric Main. 



535 



X. 



Kelvin. 



Mascart & 

 Joubert. 



Formula (32). 



5 



0-8172 





0-8172 



5-5 



0-8069 





0-8064 



6 



0-7979 





0-7976 



8 



0-7739 





0-7737 



10 



0-7588 



0-7596 



0-7597 



15 



0-7431 



0-7416 



0-7416 



20 



0-7325 



0-7328 



0-7328 



30 





0-7251 



0-7241 



40 





0-7196 



0-7193 



50 





0-7172 



0-7172 



CO 



0-7071 





0-7071 



As the approximate formulas given above are very simple, 

 it will be seen that tables need only be constructed for 

 values of x lying between 2 and 5 or 6, although to have 

 tables of other values would doubtless be a great convenience 

 to those who have to use the formula?. 



4. Particular solution. 



We have seen that a solution of the equation 

 "d 2 i 1 ~di _ m 2 'di 

 dr 2 r ~dr a ()£ 



is i = \{mrV~i) e» tl 



— (ber mr + 1 bei m?')(cos cot + 1 sin at). 



Hence, since both the real and imaginary parts of this 

 solution must satisfy the differential equation, we see that a 

 particular solution may be written in either of the following- 

 forms : 



i = (A ber mr 4- B bei mr) cos at 



+ ( — A bei mr + B bor mr) sin at, . . (33) 



0r i = (A 2 + B 2 )V2(ber 2 ttir+bei 2 mr) 1 2 cos (orf-e), . (34) 



where A and B are constants, and 



tan e = ( — A bei mr-{- B ber mr)/(A ber mr + B bei mr). 



For a solid core and an infinitely thin return conductor of 

 infinite conductivity this solution suffices, and is the one 



