A PARADOX IN THK THEORY. OF ATTRACTION. 



Let us neat consider a disk on which two chords are drawn intersecting 

 t th* point P at a small angle 0, and let corresponding elements be taken 

 of the two sectors so formed. 



In this case the section of either sector is proportional to the distance of 



lenient from the point of intersection, and therefore the two sections are 

 proportional to the values of y at the two elements. Hence if pif~ n is constant. 



particle at the point of intersection will be in equilibrium. 



If the edge of the disk is the ellipse whose equation is 



and if at any point within it 



l-^-=|>' (15), 



rt ft 



iuil if the length of a diameter parallel to the given chord is 'Zd, then the 

 value of y for any point of the chord is 



y=pd... .'.... (IG). 



Henre if p=Cp"- i (17), 



u particle placed at any point of the disk will be in equilibrium under the 

 .u-tion of any pair of sectors formed by chords intersecting at that point, and 

 then-fore it will be absolutely in equilibrium. 



When as in the case of electricity, n = 2, 



P = Cp~*. (18), 



the known law of distribution of density. 



If the repulsion were inversely as the distance, the fluid would be accu- 

 mulated in the circumference of the disk, leaving the rest entirely empty. 



If the force were inversely as the cube of the distance, the density would 

 l>e uniform over the surface of the disk. 



Lastly, let us consider a solid ellipsoid, the equation of the surface being 



-* 1 v -0 

 



and at any point within it let 



a 



