376 MAGNETIC INDUCTION. 



or, replacing M and by their values, 



M' 2 = < 2 M 2 (i - cos 2 0) = *A 4 [H 2 - (k\ + k'X" 2 + /T 2 A" 2 ) 2 ] 



= i?<? [~(/ 2 A 2 + ' 2 A' 2 + " 2 A" 2 ) ( A 2 + A' 2 + A" 2 ) - (A 2 + k' A' 2 + "A" 2 ) 

 = ty* \ [X'X'(k' - k")J + [ A" A (k" - k)J + [XX' (k - k') 



The sphere can only be in equilibrium if the couple produced by 

 the action of the field is null ; hence the three squares comprised 

 within the brackets must be separately null. As, by hypothesis, the 

 co-efficients k, k' k" are different, two of the cosines A, A', X" must be 

 equal to zero, and therefore one of the principal axes of magneti- 

 sation must coincide with the direction of the field. 



The preceding calculation applies also to the case of a homo- 

 geneous body of any given shape situated in a uniform field, for the 

 magnetic moment, in reference to one of the principal axes, is simply 

 proportional to the volume of the body, and to the component of the 

 force of the field. 



If the field is variable, we may suppose the volume of the body 

 under consideration to be infinitely small ; the moment of the couple 

 which tends to turn it about its centre of gravity will still have the 

 same expression as a function of the strength of the field at the point 

 occupied by the element of volume. 



393. EXPERIMENTAL DETERMINATION OF THE COEFFICIENTS 

 OF MAGNETISATION. When a cylinder is magnetised in a manner 

 uniformly parallel with the axis, the action which it exerts on a point 

 in the interior only depends on the two terminal layers. If the cylin- 

 der is very long, this action may be neglected for all points whose 

 distance from one end is very great compared with the diameter ; 

 the resultant force will then be produced by external masses alone. 

 If the external field is uniform and parallel to the axis, the magneti- 

 sation in the greater part of the cylinder will be uniform and 

 proportional to the strength of the field. In the neighbourhood of 

 the ends only, the induced magnetisation will be slightly modified ; 

 the superficial layer, instead of being uniform, and limited to the 

 terminal surface, will have a more complicated distribution, and will 

 be partially spread on the lateral surfaces. 



According to this the coefficient of magnetisation of an isotropic 

 substance may be defined as the quotient, by the force of the field, 

 of the intensity of magnetisation which an infinitely thin cylinder of 

 the substance acquires when placed in a uniform field; or the 

 magnetisation which it assumes in a field equal to unity. 



