384 Lord Rayleigh on the Self-induction 



rings. For an ellipse of semi-axes a and b Prof. J. J. Thomson 



gives* 



lo g R=log^-i, (1) 



in which of course the case of the circle is included. 



It is evident that for a given area R is least when the figure 

 is circular. In that case we have 



R=-4393 i/(A) (2) 



In the case of the square, 



R= -44705 •(A) (3) 



For the ellipse, Prof. Thomson's result leads to 



R = -4393V'A.(1+ T fee 4 + ...), . . . (4) 



showing the small effect of moderate eccentricity when the 

 area is given. 



As examples of very elongated forms we may take the 

 ellipse or the rectangle. In the latter case the value approxi- 

 mates to that applicable to a line given by Maxwell (5) § 692. 

 If the length be a, 



& A3 



K=zae *= — _ — ? (5) 



increasing without limit for a given area as b decreases. 



It has been pointed out that Maxwell's result (22) § 685 is 

 not rigorous, unless yu, be constant. In order to put a case in 

 which the lines of induction follow a simple law in spite of 

 the presence of iron, we may suppose that the conductors are 

 co-axal cylindrical shells. The outgoing current of total 

 strength travels in the interior cylinder of radii <2 2 , % ; the 

 return current of strength — C in the outer cylinder of radii 

 a/, a{ 



In Maxwell's notation we have the equations 



m 



dr 



= —p/3, /3r = 4t7r l iv r dr, 



so that 



( H wdxdy=2Tr j H w r dr=\ \ H d(/3r) 



Now /3r vanishes both at zero and infinity, so that we may 

 * In a private letter. 



