300 F. B. Taylor — Determinate Orbital Stability: 



being eliminated. So strong are the grounds for this 

 conclusion that it may be affirmed positively that no third 

 alternative exists; the Moon's stability must be either 

 determinate or indeterminate. 



The Mechanism of Determinate Orbital Stability. 



In discussing the Moon's motion and stability, it is 

 desirable for the present purpose to do it in the simplest 

 possible way. Let it be supposed that both the Earth 

 and the Moon revolve in circular orbits around their 

 primaries, and that the plane of the Moon's geocentric 

 orbit is coincident with the ecliptic. Then, while the 

 Moon revolves in a circle around the Earth, the Earth 

 itself revolves in a circle around the Sun, so that the 

 Moon's true path around the Sun is an epicycle. Plotted 

 in a diagram, this path is a wavy line passing alternately 

 inside and outside of the Earth's orbit, and crossing it 

 twice in each period. The assumption of these simplified 

 conditions will in no way invalidate the conclusions 

 reached. 



Upon careful examination, it was found that the 

 Newtonian or current analysis of the Moon's motion does 

 not show the Moon's orbital stability to be determinate; 

 in fact, it shows stability to be indeterminate. It was 

 necessary, therefore, to devise some other method of 

 analysis. The method emplo^^ed here has grown natur- 

 ally out of the study of determinate stability and the 

 elements which enter into it, and, in general outline, it is 

 believed to be the best if not the only method by which 

 the mechanism can be correctly analyzed. This method 

 proceeds by a direct analysis of the forces of the epicycle, 

 i. e., by a direct study of the forces which affect the Moon 

 as it moves along its epicyclic path around the Sun, and 

 avoids the customary assumptions of the Newtonian 

 analysis by which the heliocentric motion of the Earth- 

 Moon system is transferred to the Sun regarded as a 

 distant satellite, and by which the motions of the Moon 

 and the Sun are referred to the Earth as to a relative 

 center. (See "Outlines of Astronomy." Bv Sir John 

 Herschel. Edition of 1876, page 415, section (610).) 



For every possible circular orbit around the Earth, 

 there is, theoretically, a corresponding epicycle, and since 

 in each case the geocentric circle is simpl}^ the epicycle 

 with the heliocentric motion eliminated, the values for 



