Lines of Force oj a Circular Current. 29 



The table shows that the error in M calculated by the ap- 

 proximate formula (25) is only O'OOl per cent, for y = 45°, 

 and 0-09 per cent, for y = 70^. When the coils are near 

 each other the approximation can be carried still further by 

 using (27) ; thus dispensing with Legendre's tables of elliptic 

 integrals. It will not be out of place to giye a numerical 

 value for a single instance in order to show the rapid con- 

 vergence of (23). For k= sin 70° 



^ = 0-1309845 (= g Vo-0000771^ =2(^^-1 ) 



+ O-0OO0OO2( = 15(|^') 

 = 0-1310618, 



M 



log^ 7= =l-7754242(= W 4776^1) + 0-O003758(=loo- (1 + e)) 



= 1-7758000, 



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

 be noticed that the above is the most unfavourable case where 

 (24) may be applied. 



9. 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 formula already obtained. 

 The potential of a magnetic shell (f> is given by (A) 



»- 



By expanding Ji(Xa) according to ascending powers of Xa, 

 and remembering that 



= 1.2.3.. {2m + 1) (^2 + z')->n-.ii^,,,,^^(^ V^^y 



where P2OT+1 denotes zonal harmonics of 2m-\-l-xh. order, we 

 arrive at the ordinary expression for the solid angle in 

 terms of spherical harmonics. M can be similarly expressed 

 by using (B'). 



It is needless to remark that such expressions converge 

 very slowly. What I want to show in the present paper is 

 that we may sometimes arrive at a convenient and practical 

 result by using the ^-series, instead of falling into the grooves 

 of spherical harmonics. 



