160 HEXKY A. EOWLAXD 



Maxwell's formulae are only adapted to coils of small section. Hence 

 we must investigate a new formula. 13 



Let N be the total number of windings in the galvanometer. 

 Let R and r be the outer and inner radii of the coils. 

 Let X and x be the distances of the planes of the edges of the coils 



from the centre. 



Let a be the angle subtended by the radius of any winding at the centre. 

 Let & be the length of the radius vector drawn from the centre to the 



point where we measure the force. 

 Let 6 be the angle between this line and the axis. 

 Let c be the distance from the centre to any winding. 

 Let w be the potential of the coil at the given point. 



Then (Maxwell's 'Electricity,' Art. 695), for one winding. 



W = 2n ] 1 COS a + sin 2 a ( Q[ (a) $1 (#) 

 ( \c 



and for two coils symmetrically placed on each side of the origin, 



W = 4:r \ COS a sin 2 a ( * f ) O 2 ' (a) Q 2 (0) 



I \ * \ c 1 



where Q 2 (0), Q^(0), &c., denote zonal spherical harmonics, and Q 2 '()> 

 Q'i(a) &c., denote the differential coefficients of spherical harmonics 

 with respect to cos a. 



As the needle never makes a large angle with the plane of the coils, 

 it will be sufficient to compute only the axial component of the force, 

 which we shall call F. Let us make the first computation without 

 substitution of the limits of integration, and then afterward substitute 

 these: 



F = 



* f C^-dxdr, 



r)(X x)J J dx 



and we can write 



%*N 



&c. 



12 A formula involving the first two terms of my series, but applying only to the 

 special case of a needle in the centre of a single circle of rectangular section, is 

 given by Weber in his 'Elektrodynamische Maasbestimmungen inbesondere Wider- 

 standsmessungen,' S. 872. 



