6 2 J ELLIPTICITY OF THE EARTH. 501 



The quantity m denoting, as in Delaunay, the ratio of the mean 

 motion of the Moon to that of the Sun, it is found that the analytical 

 expressions of most of the coefficients involve negative powers of m. This 

 circumstance, which never happens in the case of the perturbations due to 

 the Sun's action, has given rise to a difficulty in some minds as to the 

 admissibility of Mr Hill's results. Mr Stockwell, in particular, in an article 

 in the twenty-ninth volume of the American Journal of Science, asserts 

 that the value given to the coefficient of the principal equation of latitude 

 leads to a manifest absurdity, and "justifies the suspicion that the entire 

 solution is erroneous." 



The difficulty thus noticed by Mr Stockwell, however, admits of an 

 easy explanation. He applies Mr Hill's formulae to a case in which they 

 are not applicable, and for which they were not intended. The form of 

 development in series adopted by Mr Hill is founded on the supposition 

 that the perturbations due to the Earth's figure which he wishes to deter- 

 mine are very small compared with those due to the action of the Sun, 

 and therefore he expressly neglects quantities which are proportional to the 

 square of the first-named perturbations. Now, in the case of our Moon, 

 which is that treated by Mr Hill, the above-mentioned supposition certainly 

 holds good, and consequently his formula? are sufficiently accurate. 



If, however, the Sun's distance from the Earth were very much greater 

 than it is, or if the Moon's distance were very much less than it actually 

 is, then the perturbations arising from the Earth's figure might be much 

 greater than those which arise from the Sun's action, and a different form 

 of development would have to be adopted. 



In this latter case it would be better to refer the motion of the 

 Moon, not to the ecliptic, but to a fundamental plane passing through the 

 line of intersection of the equator and ecliptic, and occupying a definite 

 intermediate position between those two planes. If the perturbations due 

 to the action of the Sun are much greater than those due to the Earth's 

 figure, this fundamental plane nearly coincides with the ecliptic, whereas 

 if the latter perturbations are much greater than the former, the funda- 

 mental plane nearly coincides with the equator. In Mr Hill's formula, the 

 principal term in the expression for the latitude nearly represents the dis- 

 tance of the fundamental plane from the ecliptic corresponding to the actual 

 longitude of the Moon at the time. 



A simple analytical illustration of the change of form of the coefficient 

 of this term of the latitude in different circumstances may be given. 



