154 THE ROYAL SOCIETY OF CANADA 



to decrease 6, the angle between the axes, is equal to — where W is 



de 



the mutual potential energy of the two systems. W depends upon 

 the current in the coils, upon the shape and size of the coils, upon the 

 strength of the magnets, upon the distribution of magnetism, and 

 upon the space included between the two systems and straight lines 

 joining their boundaries. It is the presence of this last factor which 

 is responsible for any difficulty in the matter, and it obviously resolves 

 itself into a question of solid angles. By expressing the required solid 

 angles in terms of Zonal Harmonics, a very convenient way of defining 

 and determining the constants of the magnetic system can be de- 

 veloped. 



Let us take M, Li, and L2, as the chief constants for a thin magnetic 

 system, where M is the magnetic moment, L\ is the "active" or 

 "effective" length, and L2 another unknown constant. It is on the 

 significance and evaluation of these quantities that we shall centre 

 our attention. In null methods which involve the simultaneous use 

 of coils of different sizes with a magnetometer, a correction factor 

 comes in, which includes these constants (see equation (7)), and in 

 direct magnetic measurements they become of fundamental import- 

 ance. M is well known, Li is frequently discussed, somewhat errone- 

 ously, under the title "equivalent length", and L2 is generally ignored 

 along with other constants of a lower order. 



If m is the pole strength at a point in a thin magnet, and / is 

 the distance of the point from the centre of the magnet, we 



have for the whole system, ilf=22m/; also L\^^^^^^^^^ /M; and 



L*2 = (22w/^)/^ xhe difference of each of Li and L. from M/v^ 



should be noted, because the latter quantity which is not required 

 in this analysis, is sometimes taken as the equivalent length and used 

 for Li and Lo in approximate formulae. The expression below shows 

 at once how these values arise. The potential, V, at a point r, 6 due 

 to a thin magnet, can be shown to be given by 



,. (22m/)Pi , (^2mF)Ps ^ {Z2ml')P, , 



V — -\- — ■ -\- -j- . . . . 



MP, ^ ML\P^ _^ ML\P, ^ ■ ^^^ 



where Pi, P3, P5, etc., are zonal harmonics for cos 6 of the orders 

 1, 3, 5, etc., respectively. 



