302 F. B. Taylor — Determinate Orbital Stability: 



second in opposition, its present velocity at that point 

 being 19 miles per second, and the curvature of the 

 Moon's path with respect to the Sun would also be sharper 

 than now in opposition. The Sun's attraction would also 

 be slightly stronger on account of a reduction of distance 

 amounting to about l/500th. 



Conversely, if the Moon were started in perfectly 

 adjusted revolution in an orbit, say, 480,000 miles from 

 the Earth its velocity of revolution around the Earth 

 would be slightly less, and its periodic time longer, pro- 

 ducing, of course, a very different epicycle from the pres- 

 ent one, its curves being longer and more gentle. With 

 respect to the Sun, the Moon's velocity would be slightly 

 less than now^ in opposition, and the curvature of its path 

 less sharp. The Sun's attraction w^ould also be slightly 

 weaker on account of the greater distance. The masses 

 of the Sun and the Earth and the Earth's distance from 

 the Sun remaining the same as now, would the Moon's 

 revolution be stable in either of these orbits? 



If stability is indeterminate the Moon's revolution 

 would be as stable in one of these orbits as in another, 

 for the distance of the orbit from the Earth would be a 

 matter of indifference. But if stability is determinate, 

 then the Moon's revolution would not be stable in either 

 the smaller or the larger of these orbits, or, in fact, in any 

 orbit other than the one in which it now revolves at a 

 distance of 240,000 miles. 



It seems clear that in the case of a body having free 

 motion in space, like the Moon^ stability can be made 

 determinate in the sense defined above only through the 

 action of opposing forces related to each other in such a 

 way that every departure of the Moon from its present 

 orbit immediately brings into action a force that tends 

 to drive it gradually back to that orbit, as the only place 

 in which the opposing forces are in equilibrium. This 

 kind of action is characteristic of a differential relation 

 of forces. The principle of differential action of forces 

 is common in many branches of mechanics, in electrical 

 science, etc., and there is no apparent reason why it may 

 not play an important part in celestial mechanics. We 

 may therefore assume, tentatively, that the place of the 

 Moon's stable orbit is controlled by an automatic differen- 

 tial mechanism — a natural one; and the only forces 

 available with which to produce a differential adjustment 



