POTENTIAL OF A SPHERICAL LAYER. 353 



Taking the expression 

 the second form of the development will give 



_j_ T -3 i a \ % T '3*5 f a \ x 4... 



369. POTENTIAL OP A SPHERICAL LAYER. Let us finally con- 

 sider any given spherical layer of radius a. The potential at an 

 external point M, at a distance r from the centre, may still be 

 expressed by the series 



(18) 



in which the coefficients depend on the law of distribution and on 

 the direction of the right line r. 



Let u be the angle of the right line r with the axis of z, / the 

 angle of the plane rz with the plane of yz ; the co-ordinates of the 

 point M are then 



z = r cos u , 



(19) y = rsin u cos /, 



x = r sin u sin /. 



Taking r, u and / as independent variables, Laplace's equation 

 gives 



3*(rV) W "SV i -&V 



r + - + coten u - + --. - = 0; 



A A 



