May 15, 1896.] 



SCIENCE. 



723 



ory of gravitation is whether all the inequali- 

 ties in the motion of the moon and the 

 planets admit of being calculated from their 

 mutual attraction. ' ' In order to answer the 

 question the astronomer must make the 

 calculations demanded by theory, giving 

 him the positions of the planets considered, 

 and then compare the calculated with the 

 observed place. ISTo complete solution has 

 ever been found even in the case of three 

 bodies, and for the case of a larger number 

 of planets no approximation to an entire 

 solution has been made. The complexity 

 of the problem is due to the fact that " the 

 forces which act upon the planets are de- 

 pendent upon their motions, and these again 

 are determined by the forces which act on 

 them." 



Many great mathematicians from New- 

 ton's time till now have given much of their 

 attention to the question of how to sur- 

 mount the dif&oulties. The success of the 

 partial solution is attested by the " marvel- 

 lous accuracy with which sun, moon and 

 planets move in their prescribed orbits." 

 Though the accuracy is marvellous, there 

 are two cases of greatest interest especially 

 demanding the attention of the mathemat- 

 ical astronomers. These two cases have to 

 do with the motions of the Moon and of 

 Mercury. 



The ' Tables of the Moon,' calculated by 

 Hansen and published in 1857 by the 

 British government, were supposed to pro- 

 vide the astronomer with the means of cal- 

 culating accurately the position of the 

 moon for a century or more. Prof. Grant, 

 in his ' History of Physical Astronomy,' 

 published in 1852, remarked: "Thus the 

 clouds which for a moment obscured the 

 Newtonian theory of gravitation have been 

 eifectually dissipated, and a fresh conquest 

 has been added to the long list of triumphs 

 which adorn its history." The agreement 

 of observed and calculated position from 

 1750 to 1850 is all that could be desired, 



but it has been found that previous to 1750 

 and after 1850 the calculations and observa- 

 tions do not agree closely enough to satisfy 

 the mathematical astronomer. Mr. Stone, 

 of Oxford, has published a table (M. IST. E. 

 'A. ^S., LII., No. 7, p. 478) showing the 

 'mean excess over observation of the 

 moon's tabular place in longitude for the 

 years 1847 to 1891, as computed from Han- 

 sen's tables.' It is therein shown that from 

 1847 to 1863 the calculated longitude dif- 

 fered from the observed by a mean annual 

 value of — 1".85 and no law of regular 

 change is apparent. Since 1863 the mean 

 annual error has increased at an average 

 rate of 0".75 per annum. The error now 

 amounts to about 20", equal to about j\-^ 

 of the moon's angular diameter. 



The lunar tables have been empirically 

 corrected by Newcomb and also by Tisse- 

 rand and at present the results are satis- 

 factory. However, gravitation seems un- 

 able to explain theoreticallj'^ the movement 

 of the moon's perigee. The mathematical 

 astronomer will no doubt triumph over the 

 new obstacle which presents itself to-day, 

 but, as Tisserand says, a beautiful discov- 

 ery remains to be made. 



Newcomb has stated that "another 

 change not entirely accounted for on the 

 theory of gravitation occurs in the motion 

 of the planet Mercury." Leverrier found 

 "that the motion of the perihelion of Mer- 

 cury is about 40" in a century greater than 

 that computed from the gravitation of the 

 other planets." He attributed this to the 

 attraction of a ' gi-oup of small planets be- 

 tween Mercury and the sun.' Newcomb, 

 in his recent work on ' Astronomical Con- 

 stants,' gives the result of an examination 

 of this hypothesis as well as of several 

 others. He concludes (1) "that there can 

 be no such non-sjnnmetrical distribution of 

 matter in the interior of the sun as would 

 produce the observed effect." (2) The hy- 

 pothesis of an intra-mercurial ring or group 



