Forces on the Motion of the Moon. 537 



The effect on the constant of nutation is relatively larger, — 

 about 0"'l to //# 2 in a constant of 9 //# 2. This, again, is not 

 excluded by the observations. To detect it with any certainty 

 we should need from five to ten periods of 18 years each. 



We must conclude, then, that a variable force with a period 

 of 270 years and a maximum value about 1/120 of that pro- 

 duced by the earth's ellipticity is not excluded by the observa- 

 tions. If such a force exists with a potential (ii + ^0/V 3 , so 

 that it may be treated as an addition to the earth's ellipticity, 

 we must conclude that the constant part of the addition is less 

 than 1/70 and the variable part less than 1/120 of that caused 

 by the earth's ellipticity. 



Either of the two forms of the hypothesis which are admis- 

 sible, an oscillation of a lunar magnetic axis about a mean 

 position in the lunar equator, or a variable moment with an 

 axis near the Innar axis of rotation, are difficult. There is, 

 however, still another effect not yet considered, and one which, 

 though secondary, is somewhat remarkable. It will be seen 

 in the next section that a minute libration of the moon can, 

 under certain circumstances, have a large, effect on its motion. 

 Xow any supposed magnetic attraction between the earth and 

 the moon will have its greatest effect in altering the direction 

 of the axes of figure of the moon. The force is the same as 

 that which would affect the precession and nutation of the 

 earth, but its effect is 80 times as great, in general, owing to 

 the earth's mass being 80 times that of the moon. The pos- 

 sibilities in this direction will be considered in § 10. 



9. The effect of the physical libration of the moorts axes. — 

 It can be shown that if we adopt the ellipticities of the moon's 

 figure as being of the order of magnitude determined by 

 observation, an oscillation within the plane of the lunar equa- 

 tor of the axes lying in that plane with a period of 270 years 

 and a coefficient of 50 " will produce the observed inequality 

 in the moon's motion. 



The physical librations of the moon are of two kinds, free 

 and forced. The forced librations, produced by the motion of 

 the moon round the earth, are for that reason powerless to 

 cause any sensible effect on its motion round the earth, — a 

 result which can be deduced from the equations of motion. 

 There are theoretically three principal free librations each 

 with its own period, dependent on the three ellipticities of the 

 principal sections of the moon's surface. The amplitudes of 

 these librations have not yet been observed and it is generally 

 assumed that they are zero or at least beyond detection by the 

 observations hitherto made. The ellipticities, nevertheless, can 

 be obtained from the forced librations and thence the periods 

 of the free librations. 



