Mechanism and Functions in Celestial Mechanics. 301 



geocentric velocity and radius in the two forms are always 

 exactly the same. Supposing the Moon to revolve around 

 the Earth in the direct order, or from west to east, the 

 several forces affecting it are most simply related at the 

 point of opposition, being there compounded, and hence at 

 their maximum values, the only important exception 

 being the Sun's angular pull when the Moon is in quadra- 

 tures. This force, in effect, augmenting the Earth's 

 attraction for the Moon, is at its maximum in quadratures 

 and disappears in syzygies. I have therefore confined 

 discussion mainly to the forces as they appear in opposi- 

 tion. 



On account of differences of distance, the Moon at the 

 point of opposition in every possible epicycle is acted 

 upon by different values of attraction both on the part 

 of the Earth and the Sun. On this account the Moon has 

 in every separate epicycle a different velocity and curva- 

 ture, not only with respect to the Earth, but also with 

 respect to the Sun. From this fact the so-called centri- 

 fugal force which depends upon velocity and curvature 



(abstract formula -^-) has a different value at the point 



of opposition in every separate epicycle, not only with 

 respect to the Earth, but with respect to the Sun also. 

 With changes of the Moon's orbital distance from the 

 Earth, the attractions of the Earth and the Sun vary at 

 different rates, and in order to maintain stable revolution, 

 the Moon must obey the forces of both bodies. Such a 

 relation suggests the possibility of adjustment like that 

 required theoretically for determinate stability. 



Suppose the Moon were started to revolve around the 

 Earth in an orbit at a distance, say, of 60,000 miles, its 

 motion being toward the tangent and at the precise veloc- 

 ity appropriate to circular revolution around the Earth 

 under its power of attraction at that distance. Under the 

 law of velocities in circular orbits (velocities in circular 

 orbits vary inversely as the square root of the distances), 

 the Moon's velocity around the Earth at that distance 

 w^ould be twice what it is now, or about one mile per 

 second, and its periodic time or month would be one- 

 eighth of the present month. The epicyclic path with 

 reference to the Sun would, of course, be very different 

 from the Moon's present epicycle. The IMoon's velocity 

 with respect to the Sun would be about 191/0 miles per 



