ON A PARADOX IN THE THEORY OP ATTRACTION. 603 



At any point of a chord drawn parallel to a diameter whose length is Id 

 the value of y is pd. 



If we consider a double cone of small angular aperture whose vertex is 

 at a given point, and whose axis is this chord, the sections at two correspond- 

 ing elements are in the ratio of the squares of the distances of the elements 

 from the given point, and therefore in the ratio of the values of p* at these 

 elements. Hence the condition to be satisfied is 



pp*~ n = C, a constant. 



If this condition be fulfilled the fluid will he in equilibrium at every point 

 of the ellipsoid. 



Ifn = 2, P=Cp-> 



is the condition of equilibrium. But if C is finite the whole mass of the fluid 

 in the ellipsoid if distributed according to this law of density would be infinite. 

 Hence if the whole quantity of fluid is finite it must be accumulated entirely 

 on the surface, and the interior will be entirely empty, as we know already. 



If the force is inversely as the fourth power of the distance the density 

 within the ellipsoid will be uniform. 



762 



