160 THE LAWS OF 



08) ...... p = 



P' being the density of the fluid on the exterior surface. 



If now we conceive a conducting sphere A whose radius is 

 a, and determine P' so that all the fluid of one kind, viz. that 

 which is redundant in this sphere, may be condensed on its 

 surface, and afterwards find 6 the radius of the interior sphere B 

 from the condition that it shall just contain all the fluid of the 

 opposite kind, it is evident that each of the fluids will be in 

 equilibrium within A, and therefore the problem originally pro- 

 posed is thus accurately solved. The reason for supposing all 

 the fluid of one name to be completely abstracted from JB, is 

 that our formulae may represent the state of permanent equi- 

 librium, for the tendency of the forces acting within the void 

 shell included between the surfaces A and B, is to abstract 

 continually the fluid of the same name as that on A's surface 

 from the sphere B. 



To prove the truth of what has just been asserted, we will 

 begin with determining the repulsion exerted by the inner sphere 

 itself, on any point p exterior to it, and situate at the distance r 

 from its centre 0. But by what Laplace has shown (Mec. CeL 

 Liv. II. No. 12) the repulsion on an exterior pointy, arising from 

 a spherical shell of which the radius is /, thickness dr and 

 centre is at will be measured by 



2-jrr'drp d_ (r + r') 3 ~ n - (r - rj^ 

 1 7i . 3 n' dr' r 



the general term of which -when expanded in an ascending series 

 of the powers of - is, 



.....- - , , 2 , 



2.3.4.5...2*+l ~ r ' r 



and the part of the required repulsion due thereto will, by sub- 

 stituting for p its value before found, become 



