Forces on the Motion of the Moon. 535 



The chief reason that the effect of magnetic attraction may 

 be capable of observation arises from the fact that the result- 

 ant force varies as the inverse fourth power of the distance, 

 while gravitation varies as the inverse square. It is in this 

 respect similar to the attraction produced by the ellipticity of 

 the earth's figure, that is, by the difference between the 

 moments of inertia about the polar and equatorial diameters 

 of the earth. For this reason it will be useful to compare the 

 magnitudes of the forces which must be invoked with that 

 produced by the earth's figure. It is, however, to be remem- 

 bered that the resolved part of the magnetic attraction in any 

 direction differs from that of the earth's figure, mainly 

 because the earth's magnetic axis is not the axis of figure. 



The potential of the magnetic attraction may with sufficient 

 approximation be expressed in the form (u a + u^)/r z where u Q 

 is a constant and u x consists of terms whose mean value for 

 all time is zero, and r is the distance between the moon and 

 earth. It is a simple matter to prove that a force due to a 

 potential ujr 3 must have an accelerative effect on the moon 

 less than 5*10" 9 of that produced by the earth's attraction on 

 the whole mass of the moon. This fact is deduced from the 

 well-known result that the motion of the perigee, as determined 

 by observation and theory, is not in error by so much as one 

 part in 5*10 5 . Now the principal effect of the earth's ellipticity 

 is 10" 7 of that produced by the main attraction of the earth. 

 Hence the effect of a term u /r 3 , here supposed to be due to 

 magnetic attraction, must be less than 1/20 of that of the 

 earth's ellipticity. Further, if the inequality which we seek 

 to explain is produced by a force of the same period as that 

 of the inequality, this force, contained in ujr^^ will not be 

 greater than 1/120 of that of the earth's ellipticity. 



These facts give the first negative result, namely, that the 

 inequality vie seek to explain cannot be produced solely by the 

 secular motion of the magnetic axis of the earth. This secu- 

 lar motion is known with an accuracy for the last 300 years 

 which is sufficient to prove that if it produced the inequality, 

 it would cause an addition to the mean motion of the lunar 

 perigee much greater than the agreement between gravitational 

 theory and observation permits. Such a hypothesis appeared 

 not unreasonable at one time, since there was some reason to 

 think that the period of the secular variation was about 600 

 years and this would produce an inequality in the motion of 

 the moon of half the period of the change. The period of 

 the secular change, if a real period exists, is now thought to 

 be considerably longer than 600 years. 



Next, suppose that the inequality is produced by a variation 

 in position only of an assumed magnetic axis in the moon. 



