Meclianism and Functions in Celestial Mechanics. 303 



are gravity and inertia, the centripetal and the so-called 

 centrifugal forces. Supposing stability to be determin- 

 ate, it is fair to assume, experimentally, that the Moon's 

 present distance from the Earth is maintained at 240,000 

 miles by a differential adjustment in which the centripetal 

 and centrifugal forces are, on the whole, exactly balanced 

 against each other. 



At the point of opposition in the epicycle, the Earth and 

 the Sun pull together and pull the Moon toward the Sun, 

 so that the combined centripetal forces are at the moment 

 at their maximum value, and are, in effect, heliocentric 

 forces. The velocity and curvature of the Moon's motion 

 with reference to the Sun are also at their greatest value 

 at this same moment, and the inertia with which the 

 Moon resists the deflecting force of the combined attrac- 

 tions naturally produces a like increase of centrifugal 

 force. The combined centrifugal forces are, therefore, 

 at their maximum value at the same moment, and since 

 they are merely responsive forces brought into existence 

 by the action of the heliocentric centripetal forces, their 

 action must vary in the same order and concurrently with 

 those forces. At 240,000 miles the forces are, on the whole, 

 exactly balanced against each other and stability is 

 assured. In an orbit at 60,000 miles, both the centripetal 

 and centrifugal forces would be largely increased. But 

 would they be increased by exactly the same amounts, and 

 would they show the same balanced relation that they 

 show in the present stable orbit at 240,000 miles ? One can 

 imagine a case in which the Moon might be made to con- 

 tract its orbit gradually from its present place to an orbit 

 at 60,000 miles. We should then see the two forces just 

 mentioned increasing gradually from their present values 

 to the greater values they would have in the smaller orbit. 

 Would the two forces increase at precisely the same rate, 

 or would they increase at slightly different rates? This, 

 it seems to me, is the vital point in the problem of stabil- 

 ity. Up to the present time it seems to have been assumed 

 by every one that these forces must always vary at pre- 

 cisely the same rate; but is there any inherent reason 

 why this should be so ? And if they do not increase at the 

 same rate which one increases at the higher rate? 



If the two forces in the differential mechanism 

 increased at precisely the same rate there would be no 

 reason whv the Moon should be unstable in the smaller 



