Forces on the Motion of the Moon. 533 



The hypothesis under examination demands that the sun be 

 not symmetrical with respect to every plane which passes 

 through its axis of rotation, and further, that the period of 

 rotation approximates closely to one of the lunar months. To 

 justify the former we must suppose that the nucleus of the 

 sun's mass is a solid body or behaves like one ; otherwise it is 

 improbable that an equatorial ellipticity could permanently 

 subsist. A coincidence of the period of rotation with one of 

 the lunar months is less difficult. Although the photosphere 

 shows periods of rotation differing with different latitudes, it 

 is probable that the nucleus of the sun's mass has a period of 

 rotation like that of a solid body, and that the period is some- 

 where between the greatest and the least of those observed by 

 watching the motions of sun-spots. Now this period, between 

 25 and 30 days, may possibly coincide very nearly with one of 

 the lunar months. If it does so with a difference which is 

 small enough, the effect on the moon's motion will be an 

 inequality of very long period, provided the sun has some 

 equatorial ellipticity of figure. We have, therefore, to make 

 two assumptions ; one, that the period of rotation of the sun 

 very nearly coincides with one of the lunar months, and 

 second, that its moments of inertia about two axes lying in the 

 equatorial plane of the sun are not equal. The computation 

 of such an effect is somewhat troublesome, but it gives a per- 

 fectly definite result. If the period of the sun's rotation with 

 respect to a line joining the earth and sun differs from the 

 period of the synodic month by a small quantity, so that the 

 two phases only coincide once in 2Y0 years, it is only necessary 

 to assume that the effective ellipticity* of the sun in its 

 equatorial plane is about 1/46,000. If we assume that the near 

 coincidence is with an anomalistic month, then the ellipticity 

 must be about 1,220 to produce the required term. If the 

 near coincidence is with a nodal month an ellipticity of 1/2200 

 is sufficient. 



These minute values are mainly due to the presence of the 

 factor 3600 2 , explained in §5. The introduction of two new 

 hypotheses in order to explain a single phenomenon is objec- 

 tionable and therefore requires some examination into their 

 probability. Although observation has not yet reached the 

 degree of accuracy required to compare the periods, it will be 

 advisable to examine the material in order to see how closely 

 we can compare them. 



The periods of the three months relative to a line joining 

 the mean positions of the earth and sun are : 



* The effective ellipticity is here defined to be the ratio of the difference 

 of the moments of inertia to the product of the mass of the sun and the 

 square of the radius of its visible surface. 



