SHELL WITHOUT ATTRACTION ON A PARTICLE WITHIN IT. 97 



centre of the sphere, D P C E the diameter through P, and 

 APB perpendicular to CD. In Newton's proof of the 



proposition that, if the law of attraction be that of the in- 

 verse square, the force at P is zero, the surface is divided 

 into an indefinitely great number of opposite elements by 

 small cones having their vertices at P, and the attractions 

 of each of these pairs of elements are shown to balance 

 each other. We shall first show that if the attraction at 

 P is zero, then it follows inversely that, for at least one 

 position (if not for all positions) of the coneMPm besidcvs 

 the position APB, the attractions of the opposite elements 

 balance each other; and we shall thence prove that the law 

 of attraction must be that of the inverse square. 



Let us suppose the cone, with vertex at P, to move round 

 from the position where A B is its axis to any other position 

 MPm. At AB the attractions of the opposite sections on 

 P are equal, whatever the law of the force. As the cone 

 leaves AB let us suppose the resultant attraction of the 

 two opposite elements to be no longer zero, but to act, say, 

 towards the centre side of APB. Then it will either con- 

 tinue towards that side as the cone moves all the way roiind 

 from APB to BPA, or it will vanish at some position, and 

 then act in the opposite direction. In tne first case we 

 should have a number of forces all acting irom P towards 



SER. III. VOL. VI. H 



