98 ON THE LAW OF FORCE IN A SPHERICAL SHELL. 



the same side of APB, whose resultant is zero ; then each 

 separate force must be zero. In the second case the re- 

 sultant attraction of the opposite sections vanishes some- 

 where between AP and EP ; then for at least one position 

 of the cone, besides the position APB, the resultant at- 

 traction of the opposite sections vanishes. 



Since this is true for any position of P, we can show that 

 the law of the force must be that of the inverse square. 



In the position where the two opposite sections exert 

 equal attractions, two sections of the same thickness per- 

 pendicular to the axis of the cone would also exert equal 

 attractions ; for they would bear to each other the same 

 ratio as the two oblique sections made by the sphere, since 

 these two oblique sections make equal angles with the axis 

 of the cone. Then what we have proved is, that for every 

 position of P there are two different distances for which 

 the attractions of the sections of a small cone of equal 

 thickness on a point at its vertex are equal. 



FIG. 2 



Let us take V M X to represent the axis of a cone of very 

 small angle of which V is the vertex. 



At any point M draw an ordinate MN to represent the 

 attraction of a section of the cone at M of small given 

 thickness on a point at the vertex. Then N will trace out 

 a curve as M moves along VX. 



Now take a spherical shell of thickness equal to the thick- 

 ness of the sections of the cone, and of radius nearly equal 

 to VM, where M is any arbitrary point in VX. Take a 

 point near the centre of this sphere. As a cone moves 

 round with this as vertex, its sections by the spliere must 



