POTENTIAL OF A CIRCULAR LAYER. 349 



We have thus successively 



A i . ______ 



4 ~~ ~ A*' 7W2 ~ "'"/- / ,\2 < "7wT' 



(2.4) 



A , , . 



6 2 ' ()^ 2 (2 . 4 . 6)2 ' 

 ^ 2n ~ 2 _ _j_ 



If we know the first coefficient A , all the others can be deduced. 

 This coefficient A is given by the expression of the potential on the 

 axis, which depends on the form of the layer and on the law of 

 distribution. 



The potential outside the axis is thus 



(2. 4 ) 2 ' (to* (2.4.6) 2 



For the second series, Laplace's equation would have given the 

 general condition 



which does not enable us to determine the successive coefficients in 

 the same way. 



366. In the case of a homogeneous circular layer of radius a 

 and of density equal to unity, the value of the potential P on the 

 axis, taking the centre as origin of the abscissae, is 



P 



Putting 



u = v /fl 2 + x 2 , 



which gives 



