AND THE LINES OF FORCE OF A CIRCULAR CURRENT. 15 



is only 0.001 per cent for r=45'', und 0.09 per cent for r=70". When 

 the coils are near each other, the approximation can be carried still 

 further by using (27). In all these calculations, Legendre's table of 

 elliptic integrals may be dispensed with ; n somewhat tedious opera- 

 tion lies in finding log nati — ). 



It will not be out of place to give a numerical value for a single 

 instance, in order to shew the rapid convergence of (23). For 

 fc=sin 70°, 



g' =0.1309845 (=—Z) +0,0000771 (=2^— /VW 0.0000002/ = l5(i^n 

 =0.1310618 



log ^ =1.77.54242 l = Jog 4;ra^\ + 0.0003758 (= logil + e)] 

 ATT^ax \ / \ ■ / 



= 1.7758000. 



which coincides with the value given by Maxwell. It is to be 

 noticed, that the above is the most unfavorable case in which (24) 

 may be applied. 



§ 10. It will be worth while to mention that the expression 

 for the solid angle subtended by a circle, in terms of zonal harmonics, 

 can be deduced from the formulae already obtained. 



The potential of a magnetic shell (p is given by (A). 



00 



By expanding Ji{ka) according to ascending powers of A a, 

 and rememberina" that 



