480 Relations of Electric and Magnetic Fields. 



distances EP and AE\ and the two angles (CPE and CE'A) 

 marked 6 are equal. This angle is not, however, any longer 

 the complement of the angle between AE' and the tangent 

 to the circle at E', drawn back towards P. We may 

 calculate the components of the electric field intensity at 

 P, denoting by X the component parallel to CA, and by Z 

 the component parallel to the axis of the system. 



If as before a, a' be the radii of the circles, and r be EP,. 

 so that now r 2 = a 2 + a' 2 + b 2 — 2aa' cos, <f>, the components due 

 to ad(f> at E are, as we see at once, 



,„ adcf) a' — a cos (ft ir/ -.add) 



dX e = p ■ -j- — -, aL e — pb—^. 



Thus integrating round the circle AEB we get 



\ e = 2pa\ d<j>, L e = 2pba\ -J-. (24) 



t 'o ' Jo ' 



To pass from these to the components of magnetic field 

 intensity at A due to a current of strength pa/a 1 in the 

 circle PE', we have only to interchange X and Z, and take 

 note of the directions of the components. As a little con- 

 sideration will show, we have to replace X e , which is parallel 

 to OP, by a component along PC, and then suppose that 

 turned normally out from the paper to give Z m . It will be 

 seen that the "vertical" component Z e gives a component 

 X TO parallel to EC, and that we have 



Y ,. C^co^^dcj) r/ ,i(a / -acos</))f/(f) 



X m =-2yal>] -J , L m = 2 7 « I — —- r -£_T 



(25) 



These components can be at once expressed in elliptic 

 integrals. The component Z m is the more important as it 

 enables the mutual inductance of the two circles to be found 

 by integration over the circle AEB. The mutual inductance 

 between the coaxial circles could be found by calculating 

 the total X M at P for each of a series of narrow concentric 

 rings into which the circle of radius a, say, is divided, 

 multiplying each by the area of the ring to which it belongs 

 and calculating the sum of products thus obtained. 



These reciprocal theorems of coaxial circles have not so 

 far as I know been stated explicitly before; but Sir George 

 Grftenhill has pointed out to me that in a paper in the 

 'American Journal of Mathematics,' vol. xxxix. p. 439 (1917), 

 he has given certain general reciprocal relations from which 

 they may be deduced. 



