360 Note 2: distribution of an attracting fluid 



We have in the next place to determine whether a distribution of this kind 

 is physically possible. 



Let E be the quantity of redundant fluid in the ellipsoid; then 



/. C I p"-* \Ttabc p (i - p'rfdp 

 .'a 



= 4-nabcC p n ~ 3 (i - /> 2 ) 4 dp ...... (15) 



r r 



2 



Let p be the density of the redundant fluid if it had been uniformly spread 

 through the volume of the ellipsoid, then 



...... (17) 



and if p is the actual density of the redundant fluid, 



When n is not less than 2, there is no difficulty about the interpretation of 

 this result. 



The density of the redundant fluid is everywhere positive. 

 When n = 4 it is everywhere uniform and equal to p . 



When n is greater than 4 the density is greatest at the centre and js zero at 

 the surface, that is to say, in the language of Cavendish, the matter at the 

 surface is saturated. 



When n is between 2 and 4 the density of the redundant fluid at the centre 

 is positive and it increases towards the surface. At the surface itself the density 

 becomes infinite, but the quantity collected on the surface is insensible compared 

 with the whole redundant fluid. 



When n is equal to 2, F ( J becomes infinite, and the value of p is 



zero for all points within the ellipsoid, so that the whole charge is collected on 

 the surface, and the interior parts are exactly saturated, and this we find to 

 be consistent with equilibrium. 



When n is less than 2 the integral in equation (15) becomes infinite. Hence 

 if we assume a value for C in the interior parts of the ellipsoid, we cannot 

 extend the same law of distribution to the surface without introducing an 

 infinite quantity of redundant fluid. We might therefore conclude that if the 

 quantity of redundant fluid is given, we must make C = o, and suppose the 

 redundant fluid to be all collected at the surface, and the interior to be exactly 



