524 Dr. A. Russell on the Effective Resistance 



anode and the cathode varied from 6000 volts to 10,000. 

 The current ran from 10 milliamperes to 5. A storage- 

 battery of 10,000 cells was employed. The internal diameter 

 of the discharge-tubes was 3 cm. The distance between X 

 and C (fig. 1) was 6 cm. The distance between C and Y 

 varied from 4 cm. to 10. 



Jefferson Physical Laboratory, 



Harvard University, 



Cambridge, U.S.A. 



XL VIII. The Effective Resistance and Inductance of a Con- 

 centric Main, and Methods of Computing the Ber and Bei 

 and Allied Functions. By Alexander Russell, M.A. y 

 B.Sc* 



Table of Contents. 

 I. Introduction. 



II. Mathematical formulas : — 



1. The differential equation. 



2. Kelvin's ber and bei functions. 



3. Approximate formulae for the ber and bei functions. 



4. Particular solution. 



5. The ker and kei functions. 



6. Approximate formulae for the ker and kei functions. 



7. Formulae containing both functions. 



8. The complete solution. 



III. The formulae for the effective resistance and inductance of a 



concentric main with a solid inner conductor. 



IV. Simplified formulae for particular cases : — 



1. With direct currents. 



2. With low frequency currents. 



3. With high frequency currents. 



4. W T ith very high frequency currents. 



V. The density of the current in the inner and outer conductors : — 



1. With low frequency currents. 



2. With high frequency currents. 



VI. Concentric main with hollow inner conductor. 

 VII. The impedance of a concentric main. 

 VIII. Numerical example. 

 IX. Values of m for copper conductors. 



I. Introduction. 



TN connexion with the investigation of certain phenomena 

 which occur when alternating currents of high frequency 

 flow in thick wires, a knowledge of how the effective resistances 

 and inductances of these wires vary with the frequency is 

 most important. It is of little use to have " standard " 

 inductances in high frequency circuits when we do not know 

 how their values alter with the frequency, and therefore also, 



* Communicated by the Physical Society : read January 22, 1909. 



