1877.] Theory of Attraotion. 85 



Let the distances of these points from the origin measured 



O A^ X^ P Xi Ai 



a 



in the positive direction be a^, a^, p, x^, x^, respectively, and let the 

 equation of correspondence between x^ and oc^ be 



x^-p a^-p p-x^ p-a^ 

 If x^ and x^ vary simultaneously, 



.(1). 



' (2)- 



{x^-pf {p-xf 



Hence x^ and x^ move in opposite directions, and the lengths 

 of the corresponding elements dx^ and dx^ (considered both posi- 

 tive) are as the squares of their respective distances from the 

 point P, 



If therefore AB is a uniform rod of matter attracting inversely 

 as the square of the distance, the attractions of the corresponding 

 elements on a particle at the point P will be equal and opposite. 



Now by giving values to x^, varying continuously from a^ to p, 

 we may obtain a corresponding series of values of x^, varying from 

 ttg to p, and since every corresponding pair of elements dx^ and dx,^ 

 exert equal and opposite attractions on a particle at P, we might 

 conclude that the attraction of the whole segment A^^P on a 

 particle at P is equal and opposite to that of the segment A^P on 

 the same particle. 



But it is still more evident that if A^P is the greater of the 

 two segments, and if we cut off Pa = PA^ the attractions of Pa 

 and PA^ 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 aA^, therefore the attraction of PA^ exceeds that of PA^ by a 

 finite quantity, contrary to our first conclusion. 



Hence our first conclusion is wrong, and for this reason. The 

 attractions of any two corresponding segments A^X^^ and A^X^ are 

 exactly equal, but however near the corresponding points X^ and 

 X^ approach to P, the attraction of each of the parts X^P and 

 X^P on P is infinite, but that of X^P exceeds that of X^P by a 

 constant quantity, equal to the attraction of A^a 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. 



