306 F. B. Taylor — Determinate Orbital Stability: 



tively high rate as the Moon revolves in orbits nearer and 

 nearer to the Earth. At 240,000 miles the Earth pulls 

 the Moon from the tangent 0.0535ths of an inch in one 

 second of time. In an orbit 60,000 miles from the Earth, 

 the Earth's power over the Moon is sixteen times as great, 

 and the Moon would fall from the tangent 0.856ths of an 

 inch in one second. In an orbit 15,000 miles from the 

 Earth (l/16th of the Moon's present distance), the 

 Earth's power over the Moon would be 256 times as great 

 as it is now, and the Moon would fall from the tangent 

 13.696ths inches in one second of time ; and in orbits 

 nearer to the Earth these numbers would be still greater. 

 Thus, as the Moon moves in orbits nearer to the Earth, 

 the Earth's power over the Moon increases at a much 

 higher rate than the Sun's power. Stability, and the 

 possible limitations of stability under such conditions, 

 seem to me to present a real problem. 



The Earth's velocity of motion around the Sun is now 

 about 18% miles per second, and the Moon's velocity 

 around the Earth is slightly more than half a mile per 

 second. Thus, at the point of opposition in the epicycle, 

 the Moon's velocity with reference to the Sun is now 

 about 19 miles per second, and stability is maintained with 

 this excess of heliocentric velocity as one of its conditions. 

 In an orbit at 60,000 miles, the Moon's heliocentric veloc- 

 ity in opposition would be about 191/2 miles per second, 

 and in an orbit at 15,000 miles, it would be about 211/0 

 miles per second. Along with this increase of velocity, 

 there would also be a relatively rapid increase of heliocen- 

 tric curvature in that part of the epicycle,, due to the 

 greater fall in one second of time from the tangent. The 

 amount of increase in the Earth's power over the Moon, as 

 the Moon revolves around the Earth in smaller and 

 smaller orbits, is absolutely fixed by the law of gravita- 

 tion, and we see, therefore that the Earth has no reserve 

 power by which it can increase its hold upon the Moon in 

 order to compensate a maladjustment to the Sun. In an 

 orbit of 60,000 miles radius, the Moon in obeying the 

 Earth's attraction would move too fast in opposition for 

 the Sun to do its part in holding the Moon to the curve 

 of the epicycle. But if the Sun fails the Earth must also 

 let go. This means a gradual expansion of the geocentric 

 orbit and of the epicycle, and this expansion would con- 

 tinue until the Moon reached its present place, where the 

 forces would come to a differential balance. 



