1877.] Tlieory of Attraction. 37 



In a uniform rod s is constant, so that the distribution of 

 density is given by the equation 



P=Cy--' (11). 



If w = 2, as in the case of electricity, the density is uniform. 



"We have ah'eady shown that when the density is uniform 

 a particle not at the middle of the rod cannot be in equilibrium, 

 but on the other hand any finite deviation from uniformity of 

 density would be inconsistent with equilibrium. We may there- 

 fore assert that the distribution of the fluid when in equilibrium 

 is not absolutely uniform, but is least at the middle of the rod, 

 while at the same time the deviation from uniformity is less than 

 any assignable quantity. 



If the force is independent of the distance, n=0 and 



p = Gy-' (12), 



or if r is the distance from the middle of the rod, 21 being the 

 length of the rod, 



P = f~:^ • (13)- 



If G were finite, the vvhole mass would be infinite. Hence if the 

 mass of fluid in the rod is finite it must be concentrated into two 

 equal masses and placed at the two ends of the rod. 



Let us next consider a disk on which two chords are drawn 

 intersecting at the point P at a small angle 6, and let correspond- 

 ing elements be taken of the two sectors so formed. 



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

 distance of the element from the point of intersection, and there- 

 fore the two sections are proportional to the values of y at the two 

 elements. Hence if py^'"" is constant, the particle at the point of 

 intersection will be in equilibrium. 



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



i-|-|=« ■ (»)' 



and if at any point within it 



