THE EQUILIBRIUM OF FLUIDS. 157 



contained, may be in. equilibrium in virtue of the joint action of 

 that contained in the sphere itself, and on the exterior spherical 

 surface. 



If now V represents the value of V due to the exterior 

 surface, it is clear from what Laplace has shown (Mec. Gel. 

 Liv. ii. No. 12) that 



_ _ _ . 



~)ff'^-(3-n)r [(a r) >> 



dcr being an element of this surface, and g being the distance 

 of this element from the point p to which V is supposed to 

 belong. 



If afterwards we conceive that the function V is due to the 

 fluid within the sphere itself, it is easy to prove as in the last 

 article, that in consequence of the equilibrium we must have 



V'+ V= const. 



But V and consequently V is of the form Y (0) , therefore by 

 employing the method before explained (Art. 4), we get 



where, as in the present case, j^' (0) , f^\ f^\ etc. are all con- 

 stant quantities, they have for the sake of simplicity been 

 replaced by 



J? B &c. 



Hitherto the exponent yS has remained quite arbitrary, but 

 making 

 when i = 0, 



by making /3 = - - , the formula (11) Art. 4, will become 



2.3.4 



- -n n-2.n-l...n+2t'-3 



'71-2 



sin I 



