96 APPLICATION OF THE PRELIMINARY RESULTS 



r being the distance of the point p, to which < and V belong, 

 from the shell's centre, </> (0) , < (1) , &c., U (0) ,U (l \ &c. functions of 

 6 and OT, the two other polar co-ordinates of j?, whose nature has 

 been fully explained by Laplace in the work just cited ; the 

 finite integrals extending from i = to i = co . 



If now, to prevent ambiguity, we enclose the r of equation 

 (5) Art. 15 in a parenthesis, it will become 



'da- (a 



(r) representing the distance p, do; and the integral extending 

 over both surfaces of the shell. At the inner surface we have 



r==a: nence tne P art f 



the integrals extending over the whole of the inner surface, and 

 da- being one of its elements. Effecting the integrations by the 

 formulae of Laplace (Mec. C&este, Liv. 3), we immediately obtain 

 the part ^, due to the inner surface, viz. 



In the same way the part of i/r due to the outer surface, by 

 observing that for it - = -f- and r = a l , is found to be 



dw dr 



The sum of these two expressions is the complete value of ^, 

 which, together with the values of < and V before given, being 

 substituted in the equation (c) Art. 15, we obtain 



const. = (l-g) 2<!>r+- 1 + (1 -ff) 2< (i) r l + S U t (i) r-^+ Z7<V 



