170 
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. 
Let us take VMa5 to n fig: 2 
represent the axis of a 
cone of very small angle V M x 
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 thick- 
ness on a point at the vertex. Then N will trace out a 
curve as M moves along Nx. 
Now take a spherical shell of thickness equal to the 
thickness of the sections of the cone, and of radius nearly 
equal to VM, where M is any arbitrary point in Nx. Take 
a point near the centre of this sphere. As a cone moves 
round with this as vertex, its sections by the sphere must 
be always at distances very nearly equal to the radius from 
the vertex, and by what we have proved above for some 
position of the cone the attractions of the opposite sections 
must be equal. Therefore (in fig. 2), for two distances very 
nearly equal to YM the ordinates must be equal to one 
another. Then the tangent to the curve near N must be 
parallel to Nx. But M is arbitrary, for we can take the 
sphere of any size, therefore at all points the tangent to the 
curve is parallel to ~Yx, and therefore the curve must be a 
straight line parallel to Nx, or the attractions by sections 
of the cone of equal thickness are constant wherever the 
sections be taken. But the sections are proportional to the 
