302 Prof. Minchin on the Coefficient of Self-induction 



of the wire is a (as in my paper on the " Magnetic Field close 

 to the Surface of a Wire conveying an Electrical Current," 

 Phil. Mag. August 1893). 



Then we shall calculate the total normal flux of force through 

 any surface which is intersected once in the positive direction by 

 every tube of force emanating from the given current. 



This quantity, divided by the current-strength, is the co- 

 efficient of self-induction of the current. Taking the general 

 case, viz., that in which the current-density at every point in 

 the cross-section of the wire varies inversely as the distance 

 of this point from the axis OV — we have found (Phil. Mag. 

 bid. p. 218) that at any point, P, close to the wire 



L 



Qx = 4tai J 7r — 1 — -j-coscf) |L — 1 — -; — ^ 

 \2 4a r L 4m 2 J 



1 r T /3c 2 m 2 \ m? 



15c s 



-oo.*{l£ + J-» ■-£}]}. (4) 



where w? = PD, 6= /_ PDA, L = log e — . 



The surface through which we shall take the flux of force 

 is that which is represented in section by BJEOFKA, i.e., a 

 surface consisting of the upper half portion of the anchor- 

 ring formed by the wire and of its central aperture (which 

 latter is a circle whose diameter is FE). Obviously this 

 surface is intersected by all the tubes of force. Any surface 

 starting from B and going round to A, i. e., any surface 

 having the circle of diameter AB for bounding edge, would 

 do equally well, so far as the above condition is concerned ; 

 but the calculation is simpler for the first. 



The flux through the aperture FE is, then, the value of 

 27rG . x at E, which is obtained by putting m = c, <j) = in the 

 above. 



Thus the flux through the aperture is 



w {r- 1 -s( L -i)-iM^( 2L+19 )}- -W 



To calculate the normal flux through the upper half of the 

 anchor-ring, we must take the value of the magnetic potential, 

 0, at any point close to the ring. This is the resultant conical 

 angle subtended by the circuit at the point, multiplied by i ; 

 and it is therefore (Phil. Mag. ibid. p. 213) given by the 

 equation 



