18 
This shews that the force varies inversely as the square of 
the distance. 
It will be seen on examining this demonstration that it 
involves very few properties of the ellipse, and those only of 
the most elementary kind. It introduces an important result, 
namely that involved in the relation \»=4¢. And the theory 
of limits is only required in a form which may be easily under- 
stood and admitted. 
After arriving at the result ~=4¢ we might complete the 
demonstration thus: Let PS@Q be any finite angle as before; 
let p be adjacent to P and g to Q, such that the angles P&p 
and QSgq are equal, and p and gq fall between P and Q. 
Then since the angles pSg and PSQ are bisected by the 
same straight line, the central force produces as much effect 
while the body moves from P to p as it does while the body 
moves from g to Y. But the times of describing these portions 
are ultimately as SP* to SQ’; and therefore the forces at P 
: to rel 
SP se: 
and @ are ultimately as 
ProFEssoR ADAMS made a few remarks on this communica- 
tion, describing a somewhat similar investigation which he had 
used in his own Lectures. 
