THE PROBLEM OF THREE BODIES 481 



of small motions " we may separately allow for the difference of 

 attraction which each other planet exerts upon itself and the 

 sun, which differences are known as " perturbations," till we 

 obtain as close an agreement between theory and observation 

 as we desire. In the case of the moon, though the sun (the 

 main disturbing body) attracts our satellite more than twice as 

 much as the earth does, yet since the deviations from elliptic 

 motion are due not to the whole action but only to its difference, 

 which is but a small fraction of the whole amount, similar 

 methods apply here also. 



It is, however, the opinion of high authorities that the lunar 

 motions cannot be completely accounted for by gravitation alone 

 and an important paper on the " Fluctuations of the Moon's 

 Motion " was one of the last contributions of the late Prof. New- 

 comb to astronomical science {MontJily Notices R.A.S.y January 

 1909). 



These methods, depending on a series of successive approxi- 

 mations for cases where the attraction of one body greatly 

 exceeds that of the second, on the motion of the third, cannot be 

 expected to give solutions of the general problem. 



The advance made by G. W. Hill consisted in substituting 

 the "variational curve" for the ellipse as the moon's inter- 

 mediate orbit ; thus was introduced the idea of periodic sohitions 

 into the problem of three bodies. Previous work had always 

 begun by taking the undisturbed ellipse which the moon would 

 describe under the attraction between itself and the earth alone, 

 and supposing the various elements of this orbit to be con- 

 tinually changing under the sun's disturbing action. The moon 

 was supposed to be always moving in an ellipse but the shape, 

 size and plane of the ellipse itself was in a continual state of 

 flux. At any instant, however, there will be only one ellipse, 

 which will pass exactly through the moon's position at that 

 moment and in which the velocity will be the same as the 

 actual velocity. If at this moment we suppose the disturbing 

 force to cease the attraction of the central body will cause the 

 moon to continue moving in this curve, which is known by 

 the name of the "instantaneous ellipse"; it is this curve 

 which was taken as the standard. 



Hill's " variational " curve may be described as the moon's 

 circular orbit supposed in the plane of the ecliptic, affected 

 solely by the inequality known as the "variation" (which, as far 



