528 Dr. A. Russell on the Effective Resistance 



he computed the numerical value of the effective resistance 

 in various cases. 



We may define Kelvin's functions by means of the equation 



I (mr \J l) = ber mr + i bei mr. 



Prom (1) we deduce at once that 



t . m 4 r 4 mV 8 , KN 



and 



m 2 r 2 



bei 



mr 



2 2 2' 2 . 4? . 6 2 "*~ '"' ' ' ' ' ' 



which are the definitions o£ them given by Kelvin. 



In Mascart and Joubert's V Electricite et leMagnetisme, vol. i. 

 p. 718, several of the values of the functions given in Kelvin's 

 paper have been recomputed and certain corrections made. 

 As the author uses these functions in the solutions given 

 below, it was necessary therefore to recheck the calculations. 

 He at first attempted to do this by direct calculation, but the 

 work proved so laborious that he was led to devise shorter 

 methods of calculating the functions. He found that this 

 was easy and that comparatively simple formulas can be 

 obtained for them in those cases where the direct computation 

 by (5) and (6) would be laborious. 



The formulas given below can also be usefully employed in 

 simplifying the formulse ordinarily given for computing the 

 eddy-current losses * in a metallic cylinder, the inductance 

 and resistance of two parallel cylindrical conductors f, the 

 impedance of a solenoid with a cylindrical metal core J, &c. 



3. Approximate formula for the ber and bei functions. 



When the argument is small the functions can be readily 

 computed from the formulae (5) and (6). These functions 

 generally occur associated together in one or other of the 

 following ways : — 



XW =ber 2 .r + bei 2 ^ Y(x) = ber' 2 x + bei' 2 #, 



Z(V) = ber x ber' x -f bei x bei 7 x 7 



and W(V) = ber x bei' x — bei x ber' x. 



In these definitions of X, Y, Z and W, ber 7 x and bei' x 

 stand for the differential coefficients of ber x and bei^ with 



* A. Russell, ' Alternating Currents/ vol i. p. 374. 

 + J. W. Nicholson, Phil. Mag. [6] xvii. p. 255, 1909. 

 X 0. Heaviside, ' Electrical Papers,' vols. i. and ii., or R. T. Wells, 

 JPhys. Rev. xxvi. p. 357 (1908). 



