OK A PARADOX IN THE THEORY OF ATTRACTION. 



Bat it is still more evident that if A,P is the greater of the two segments, 

 and if we cut off /*a-/M, the attractions of Pa and PA l on the particle at 

 P will be equal and opposite. But the attraction of PA, exceeds that of Pa 

 by the attraction of the part o.-!,, therefore the attraction of PA, exceeds 

 that of PA t by a finite quantity, contrary to our first conclusion. 



BPM* our first conclusion is wrong, and for this reason. The attractions 

 of any two corresponding segments yl,A', and J,A', are exactly equal, but 

 however near the corresponding points A', and A', approach to P, the attraction 

 of each of the parts X,P and A',/' on P is infinite, but that of X 3 P exceeds 

 that of A',/* by a constant quantity, equal to the attraction of Ajt on P. 



This method of corresponding elements leads to a very simple investigation 

 of the distribution on straight lines, circular and elliptic disks and solid spheres 

 and ellipsoids of fluids repelling according to any power of the distance. 



The problem has been already solved by Green* in a far more general 

 manner, but at the same time by a far more intricate method 



We have, as before, for corresponding values of x l and #, 



-1 --- L = _I ___ L. ........................ (I) . 



Z,-p 0^-p p-X, Jp-0, 



Transposing -- 1 -- = -- f- - (o) 



x,-p p-a, p-x, a,-p 



Multiplying 



If we write -xx-a = - (4), 



(5), 



we find from equation (3) ^ = P~ X * (6] 



yi y, 



Let /> p, be the densities and .<? *, the sections of the rod at the 

 corresponding points A', and X 9 and let the repulsion of the matter of the 



Ooorge Green. " Mathematical Investigations concerning the laws of the Equilibrium of Fluids 

 nalogotu to the electric fluid, with other similar researches." Transactions of the Cambridge Philosophical 

 (Read Nov. 12, 1832.) Ferrers' Edition of Green's Papers, p. 119. 



