354 PARTICULAR CASES. 



from which follows the general condition for the coefficients 



n(n + i)A n + -^ + cotan u ^ + -^- . -2 = 0. 



' n 2 2 



The general integral of this equation was given by Laplace ; if we 

 put 



_f n-m (n-m}(n-m-i} n-^-2 

 L 2( 2 -i) 



Isin^ 



(\ / \ 



2H I/ (271 2) 



the coefficient A n , expressed by means of the new symbols A w . m , 

 consists of 2 + 1 terms developed according to the sines and 

 cosines of multiples of the angles /, and its value is 



The factors denoted by g, h> with the different indices, are 

 numerical coefficients which must be determined in each special 

 case. 



370. If we consider a sphere magnetised in any given way, its 

 external action is equal to that of two layers of equal mass and 

 contrary signs, distributed on the surface according to a certain law. 



The coefficient A of the first term is null. For, in fact, at a 

 great distance, the potential simply becomes equal to the quotient of 

 the total -mass by the distance. The product A # 2 which forms the 

 numerator of the first term represents in this case the total mass 

 and we know that in every magnet the total mass is null. 



The value of the coefficient A x of the next term is 



or, taking equations (19) into account, 



