352 PARTICULAR CASES. 



in which A , A] . . . B , Bj . . . are constants, and X 1? X 2 . . . functions 

 of the angles known as Legendre's polynomials, and which are 

 denned by the series 



- 2x cos e + x *\ = 



all these functions become equal to unity when the angle is equal 

 to zero. 



As we know the development of the potential of a homogeneous 

 circular layer for a point of the axis that is to say, when is zero 

 and r equal to x, the coefficients are known. It follows that the 

 potential P for a point outside the axis is expressed by 



[r i />\* i i /r\ 4 i i -\ /A 6 "1 

 I -x l -+-xJ-}- x 4 (-}+- -|x 6 (-) -.. , 

 1 a 2 2 \aJ 2.2 *\aj 2.4.6 *\a J 



A 7 



- ) + . . 



r J 



P 



P = 27T<2 ---- 



2 



2.4 r 2.4.6 r 2.4.6.8 



368. POTENTIAL OF A UNIFORM CIRCULAR SHELL. The 

 potential of a uniform circular shell may now be obtained by the 

 expression 



We thus find, with the first form (12), 



The first terms of the development are then 



(16) 



V = 



and the series is convergent whenever p< u. 



