Electrostatic and Magnetic Fields. 479 



<#! for sin -1 (a/a'), 



CdO _ pi d6 



i f*> 



= >2 2 \ MW cos + (a 2 -a /2 sin 2 0)4}, 

 or again, since a / sin0 1 =a, 



where ^= sin" 1 {(a'/a) sin 6}. 



The second part of the integral \d0/r for the semicircle 

 •is found similarly to be 



Tims the total value of \dO/r for the circle is 

 4a'/j 1 2 C%* cos 2 ^o% 

 a' 2 -a 2 L (l-^sm'Y)* 



.{VKCAO + E^O-K^)}, (22) 



Aa f 

 a!' -a 

 where (with a' >a) k 1 2 = a 2 /a' 2 . Also 



k=Waa'l{a + a r ) = 2\/^/ (1 + *i). 



Now k and & x are the moduli connected by Landen's 

 transformation, and so the magnetic field intensity at A due 

 to a current 7 in the circle of radius a' is, as stated in § 1 

 above, 



F„,= ^ 2 E(/,) = 2 7 f^ + ^l}.. (23, 

 a w — 6r La —a a t-a) v 



A comparison of the reduced form of (14), from which (15) 

 was obtained by assuming (16), with (23) establishes (16), 

 and so evaluates this simple form of the elliptic integral of 

 the third kind. 



5. Field intensities not in the plane of the circular 

 distributions. 



Returning to fig. 1 we may regard the points E' and P as 

 in a plane parallel to that of the circle AEB, and distant b 

 from it, so that the two circles are now coaxial. The two 



