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LV . On the Self-induction and on the Gravity -Potential of a 

 Ring. By W. M. Hicks*. 



THE self-induction of a finite ring has been considered by- 

 Professor Minchin in the March number of the Philo- 

 sophical Magazine on the assumption that it is the flux 

 through a diaphragm composed of the aperture and half the 

 surface of the ring. In reading this paper it occurred to me 

 that the problem could more naturally be attacked by means 

 of Toroidal Functions. In considering it from this point of 

 view I was led to a conclusion which does not seem to have 

 been heretofore noticed. 



The expression for the self-induction of a thin circular ring 

 as given by Maxwell is 7rR(41og 8R/? 1 — 8), and for a ring of 

 finite size as given by Minchin is 7rR(41og 8R/V — 8) + terms 

 involving the radius {r) of the transverse section. In the 

 finite ring the current-density varies inversely as the distance 

 from the straight f axis. In Maxwell's wire the distribution 

 of current across the transverse section is considered uni- 

 form, and it would therefore appear that his formula gives 

 the self-induction so far as it is independent of the thick- 

 ness, and that this part is independent of the distribution of 

 current-density within the wire itself. As a matter of fact, 

 however, the distribution of current-density has a deciding 

 effect on the value of the second term, and this however small 

 the section of the ring may be. The correct value of the self- 

 induction, neglecting powers of r/R, is 7rR(4log 8R/> — 7). 

 The expression 7rR(4log8R/r — 8) gives the value of the flux 

 through the aperture only, which to this degree of approxi- 

 mation is independent of the law of current-density. The 

 difference is due to the fact that although the section across 

 the ring may be exceedingly small, yet the forces are cor- 

 respondingly large, and the interaction of the different parts 

 of the current cannot be neglected even when the ring 

 is extremely thin and of large aperture. E. g. in a ring of 

 1 metre radius and 1 millim. thick the usually accepted 

 formula gives a value 3 \ per cent, too small. 



In the present paper the question is treated by the method 

 usually employed in dealing with problems of electric potential 

 and fluid motion, but which, so far as I know, is new in its 

 application to electromagnetic and gravity potentials. The 



* Communicated by the Author. 



t By straight axis is meant the line through the centre of the ring 

 perpendicular to its plane. The circular axis is the locus of the centres 

 of the transverse sections of the ring. 





