ON A PARADOX IN THE THEORY OF ATTRACTION. 601 



rod vary inversely as the w th power of the distance, then the condition of 

 equilibrium of a particle at P under the action of the elements dx l and dx 3 is 



p 1 s l dx 1 (x 1 -p)~ n = -p^jlx^p-x,)-* ..................... (7). 



Eliminating dx 1 and cfax, by means of equation (2), we find 



p 1 s 1 (x 1 -p)*- = pA (p- x ,)*- ........................ (8), 



and from this by means of equation (6) we obtain 



as the condition of equilibrium between the elements. 



The condition of equilibrium is therefore satisfied for every pair of elements 



by making 



psy-~ n constant = C ........................... (10). 



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

 given by the equation 



P=Cy-"- .................................... (11). 



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



We have already shewn 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 therefore 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 = and 



P=0r* .................................... (12), 



or if r is the distance from the middle of the rod, 2l being the length of the rod, 



If C were finite, the whole 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. 



VOL. n. 76 



