350 PARTICULAR CASES. 



we have thus, for the first development of the potential as a function 

 of powers of /a, 



A =2Tr(u-x), 



<>A flu \ (* \ 



= 27T( - I )=27r( I ), 



^ V x J \ u 

 "^ = 2Ir w ' etc ' 



The potential at P outside the axis is 

 = 2 



The successive differentials would be easily calculated. 



367. When the layer is circular, it is often more advantageous to 

 carry out the development in another manner. 



Let a be the radius of the circle which bounds the layer, r the 

 distance of the point M from the centre of the circle, and 6 the 

 angle which the direction of this right line makes with the axis. We 

 may express the potential by one of two series 



V <$/ 



according as r is smaller or larger than a that is to say, that the 

 point M is inside or outside the sphere of radius a. The coefficients 

 are functions of the angle 0, and as the two expressions should have 

 the same value on the sphere, they satisfy the condition 



AO+AJ+ ..... = B O + B I + ...... 



The potential being considered as a function of r and of 6, 

 Laplace's equation becomes 



OT DP W VP 



r 2 -+ 2 r + cotan 6 4 = . 



or 2 or ov o# 2 



