536 Brown — Effects of Magnetic and Gravitational 



In an exactly similar manner, it is proved that this axis can- 

 not oscillate about a mean position coinciding with or near 

 the moon's axis of rotation. The only way in which the varia- 

 tion of position is effective is an oscillation about a mean 

 position in the lunar equator. Further, if the oscillations are 

 perpendicular to the equator they must be large (of the order 

 of 20°). If the oscillations are within the equator, the product 

 of the magnetic moment and the amplitude of the oscillation 

 becomes a factor which can be determined from the observed 

 coefficient of the inequality, but there is no information at 

 hand to separate the two terms of the product, since a motion 

 of the perigee is not produced to this order of magnitude. 



Let us next suppose that the directions of the magnetic axes 

 are fixed with respect to the axes of figure of the two bodies, 

 and that the inequality is produced by a variation of the 

 product of the magnetic moments. Then, owing to the facts 

 previously mentioned concerning the possible magnitudes of 

 the additional motion of the perigee and the variable force, 

 the extreme range of values of the magnetic moment is never 

 nearer to unity than is the ratio 5/7. That is, the maximum 

 variation from the mean is at least 15 per cent, and its period 

 must be in the neighborhood of 270 years. 



A magnetic attraction between the earth and moon has, how- 

 ever, a relatively greater effect on the precession and nutation 

 of the earth than on the motion of the center of the moon, 

 since the principal part is a couple, and the force we have been 

 considering is small compared with this couple. An idea of 

 the maximum magnitude of the assumed forces can be thus 

 obtained. The ellipticity of the earth's surface, as measured 

 by Clarke, is 1/293. The formulae of precession and nutation, 

 combined with an assumed equipotential surface for the earth, 

 give an ellipticity to the latter of 1/297. The observed coeffi- 

 cient of the nodal term in the motion of the moon gives an 

 ellipticity of a similar order of magnitude. The difference 

 between the two values quoted amounts to about 1/70 of the 

 whole. This limits the resolved part of the magnetic moment 

 along the lunar axis of rotation to 1/70 of the earth's ellip- 

 ticity. Hence u /r s must be less than one-half of ujr*. 



The formulae of precession compared with observation do 

 not exclude such a variable part. The amount of the preces- 

 sion in 70 years is 3600*, and the maximum change in this 

 time due to a variable force of the magnitude considered is 

 about 2". The degree of accuracy attained by observation 

 does not exclude such a coefficient. It might even be possible 

 to detect it, but the investigation would have to extend over 

 the whole range of known observations, and a very large num- 

 ber would be required. 



