[From the Proceedings of the Cambridge Philosophical Society, Vol. III. 1877.] 



LXXXV. On a Paradox in the Theory of Attraction. 



LET A^AZ be a straight line, P a point in the same, X lt X 3 corresponding points 

 in the segments PA U PA y 



Let the distances of these points from the origin measured in the positive 



A S P X A 



a 



direction be a lt a 2 , p, x lf x. 2 , respectively, and let the equation of correspondence 

 between a;, and x t be 



JL 1 J_ J_ m 



V 1 /' 



If x l and x a vary simultaneously, 



ClX-t CtJCq , \ 



T \i = 7 s (2). 



i 'Y* .I. i /n l" / <n ^ o" i 



I ut-j ^^ /y I i /_/ iX^o / 



Hence o^ and j move in opposite directions, and the lengths of the 

 corresponding elements dx l and dx^ (considered both positive) 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 lt varying continuously from a to p, we may obtain 

 a corresponding series of values of x. 2 , varying from to p, and since every 

 corresponding pair of elements dx l and dx a exert equal and opposite attractions 

 on a particle at P, we might conclude that the attraction of the whole segment 

 AJ* on a particle at P is equal and opposite to that of the segment Af on the 

 same particle. 



