NEWTON S PRINCIPIA. 141 



for the results are easily enough obtained, and in con 

 venient forms. 



If A E B is the sphere whose attraction upon the point 

 P it is required to determine, whatever be the proportion 

 according to which that attraction varies with the distance, 

 and only supposing equal particles of A E B to have equal 

 attractive forces ; then from any point E describe the circle 

 E F, and another e f infinitely near, and draw E D, e d 

 ordinates to the diameter A B. The sphere is composed 

 of small concentric hollow spheres E e f F ; and its whole 

 attraction is equal to the sum of their attractions. Now 

 that attraction of E e f F is proportional to its surface 

 multiplied by F /, and the angle D E r being equal to 

 D P E (because P E r is a right angle by the property 



of the circle), therefore E ; = - - , and if we 



call P E, or P F = r, E D = y, and D F = x, D d will be 

 dx y and Er = - ; and the ring generated by the re- 



*/ 



volution of r E is equal to r E x ED, or r E xy; therefore 

 this ring is equal to rdx, or the attraction proportional to 

 the whole ring E e will be proportional to the sum of all the 

 rectangles P D x D d, or (a x)dx , that is, to the integral 



2 ax _ x z 

 of this quantity, or to ^ - ; which by the property of 



y* 



the circle is equal to - Therefore the attraction of the 



solid E efF will be as ?/ 2 x F /, if the force of a particle 

 F/on P be given; if not, it will be as y 1 x F/x / that 

 force. Now d x : F f :: r i PS, and therefore F/ = 



PSxdx , . yxPSxrf#x/ 



- , and the attraction of L ef F is as - - - ; 

 r J r 



or taking/ = r n (as any power of the distance P E), then 

 the attraction of E efF is as P S . r n - l if dx. Take D N 



