TRANSACTIONS OF SECTION A. 615 



that subsequently the terms containing powers and products of the eccentricitiea, 

 inclination, and ratio of the parallaxes could be calculated in turn, the lower orders 

 being taken first. This point must have presented itself to Laplace and Pontdcoulant 

 in their respective theories ; but it is brought out more clearly in those treatises 

 where the object is not to obtain a complete analytical development, but to exhibit 

 the method of procedure. In Delaunay's theory this point does not present itself, 

 but by modifying Delaunay's theory so as to reduce the number of variables from 

 six to" two the variational terms might be calculated independently, whereas by no 

 process could the other terms be calculated before the variational terms. 



These considerations point to the curve indicated by the variational terms being 

 treated as the intermediary orbit in preference to the ellipse ; but it was not till 

 the year 1877 that this idea was developed. Dr. G. W. Hill then published three 

 papers in the tirst volume of the ' American Journal,' papers which Poincar^ 

 describes as containing the germ of the greater part of the progress that astronomy 

 has since made. 



Relatively to the sun's mean place the variation terms define a closed curve 

 which the moon under suitable initial conditions could describe, if the sun's 

 parallax and eccentricity were zero. The curve is symmetrical in all four quadrants, 

 and remains symmetrical about the line of syzygies, when the sun's parallax is taken 

 into account. " Dr. Hill has drawn the variation curve for ditlerent ratios of the 

 month to the year. The curve for small values of this ratio does not ditfer much from 

 a circle, but is slightly elongated in quadrature. This elongation increases with the 

 length of the month, and when there are only 1-78265 months to the year the curve 

 lias cusps on the line of quadratures. Such an orbit must certainly be unstable, and 

 by a discussion of Jacobi's equation of relative energy Dr. Hill makes it appear 

 probable that instability sets in for a much smaller value of the ratio. No numerical 

 limit has, however, been obtained for stabihty. M. Poincar(5 has succeeded in 

 obtaining the general shape of these curves when continued beyond the cusped 

 curve. The orbit first crosses the line of quadratures obliquely, and then recrosses 

 at a greater distance at right angles, then returns to the first intersection, thus 

 forming a loop, and ultimately forms a closed curve with two loops, six intersec- 

 tions with the line of quadratures and two with that of syzygies. Dr. Hill has 

 also calculared algebraically and numerically and with extreme accuracy the 

 coefficients of the diflerent periodic terms that "define the variation curve. 



In the older lunar theories the physical meaning of the quantity that in 

 ■undisturbed motion denoted the eccentricity disappeared upon the introduction of 

 the disturbance with the ellipse upon whicli it depended for its definition. At the 

 conclusion of the theories it was defined analytically by equating a given function 

 of it to the coefficient of one of the periodic "terms. Such a definition is merely 

 analytical, and has no physical interpretation. The quantity, in fact, is a mere 

 constant of integration. It is not even correct to say that it reduces to the 

 eccentricity when the sun's mean motion is put zero. The eccentricity of the 

 ellipse obtained by putting the sun's mean motion zero is a function of the constants 

 of integration and of the position of the sun's apse, and of the sun's parallax. By 

 Dr. Hill's investigations, however, a physical meaning is restored. It is a parameter 

 defining the amplitude of the oscillation that takes place about that state of steady 

 motion relatively to the sun that Dr. Hill has shown can take place, provided only 

 the sun's eccentricity be negligible. With the notion of eccentricity, the notion of 

 the apse of the older theories becomes indistinct. The so-called apse is certainly 

 not a point where the' moon is moving at right angles to its radius vector. Ac- 

 cording to Dr. Hill's theory, the period of revolution with respect to the apse now 

 becomes the period of the oscillation about steady motion. In like maimer the 

 inclination of the orbit may be considered as another oscillation about steady motion, 

 the inclination constant of integration defines its amplitude, and the period of 

 revolution with respect to the node, which isnot accurately the point of intersection 

 with the ecliptic, is now the period of this second oscillation. In accordance with 

 the general theory of small oscillations, the two modes of oscillation can co-exist in 

 complete independence so long as the squares of the amplitudes are negligible. The 

 two periods in such a case depend only on the circumstances of steady motion, in 



