NOVF.MBEK. 1911. 



KNOWLl-.DGE. 



437 



of Hansen's tables, which were an immense improvement on 

 all that preceded them ; they were considered, in the words of 

 Sir John Herschel, '" As a practical completion of the Innar 

 theory, at least for the present age. and as establishing the 

 entire dominion of the Newtonian theory and its analytical 

 application over that refractory satellite." These high 

 expectations were doomed to some disappointment ; in a few- 

 years the errors of the new tables became serious ; an 

 examination of the cause was made by Professor Newcomb ; 

 he found that the main part of the error was due to Hansen's 

 Venus terms : the one of two hundred and thirty-nine years 

 period should be omitted altogether, and that of two hundred 

 and se\enty-three years seemed to need an alteration of phase. 

 N'ewcomb further reduced the secular acceleration from 

 12"- IS to 8"-42. Hansen supposed he was using the theore- 

 tical value, and also that indicated by ancient eclipses, but 

 Adams showed that the \alue really given by theory was onl\- 

 6". and a more careful study of the eclipses showed that the\- 

 were better represented by 8" or 9". Newcomb further 

 showed that in consequence of his changes the motion of the 

 moon in a century must be reduced by 29"- 17. His 

 correction to the two hundred and seventy-three year term 

 was an empirical one (this expression denotes a term derived 

 from observation alone, without any known basis in theory). 

 It had a coefficient of 15^" and a period of two hundred and 

 seventy-three years like the Venus term. All who have 

 studied the moon's motion have found the need for such a 

 term, though they differ about its exact size and period. 

 Newcomb's corrections, which were introduced in the Nautical 

 Almanac in 18S3. were a .great improvement on Hansen : 

 the objection that some people made that thej- introduced 

 empiricism into astronomy was not a fair one ; Hansen's 

 tables were themselves empirical as regards the erroneous 

 Venus term, which has no basis in theory ; three of New- 

 comb's four corrections brought Hansen nearer theory than 

 before, only the fourth was empirical. In recent years the 

 moon's place has begun to deviate considerably from Hansen 

 corrected by Newcomb. so the need of new tables has been 

 felt. The great French astronomer Delaunay was engaged on 

 his lunar researches at the same time as Hansen, but in a 

 different manner. His idea was to find al.gebraical expressions 

 for all the terms as functions of the elements of the solar and 

 lunar orbits. The advantage is that the effect of a change in 

 the adopted value of any element can at once be deduced, 

 also that the investigation serves for other satellites, provided 

 their eccentricities and inclinations are not too great 

 (they break down for the new satellites of Jupiter 

 and Saturn). The drawback is that the expressions 

 do not converge rapidly, and that even after com- 

 puting a very large number of terms, he had reason 

 to fear that those of higher orders were still sensible. 

 Delaunay was unhappily drowned at Cherbourg in 

 1872, when the theoretical work was nearly com- 

 pleted, but the preparation of tables scarcely begun. 

 These long remained in abeyance, till the Bureau 

 des Longitudes took up the matter, and it has now 

 been concluded with the help of the late Professor 

 Tisserand, MM. Schulhof. .Andoyer and others. It 

 is not a blind following of Delaunay, whether 

 right or wrong, but advantage has been taken of 

 the work of Kadau, Brown. Cowell. and others, to 

 correct and supplement those terms that seemed to 

 be defective, if necessary carrying Delaunay's de- 

 velopments to a further approximation ; this has 

 been done chiefly by M. Andoyer. The new tables 

 diminish Hansen's mean motion by 27" per century, 

 and his acceleration by 4"-4S, making it 7"- 7. 

 Instead of Newcomb's empirical term of two 

 hundred and seventy-three years, they introduce 

 the following two empirical terms : 



of tiny planets circulating rou:.d the sun at a distance 

 about 0-178, period 27-3 days, almost the same as 

 that of the moon. If the total mass of the swarm was 

 one-fifteenth of that of Mercury, that is one-third of 

 that of the moon, there would result a term in the 

 moon's longitude with coefficient 12", period two hundred 

 and seventy years." Other positions of planets are also given 

 which might produce such a result. It is a striking instance of 

 the possible importance of small masses when their periods are 

 almost commensurable with other periods. It will probably be 

 of general interest to note the arithmetical values of the more 

 important terms in the new tables and the corresponding 

 values found by Cow-ell from an analysis of the Greenwich 

 observations : the latter is denoted by C. For the Evec- 

 tion thev use 4586"-32. C. 4586"-4; variation 2369"-95. 

 C. 2370"-2; Principal Elliptic Term 22639"-75. C. 22639"-.S; 

 .Annual Equation —658"- 92, C. —668"- 2: Parallactic inetjuality 

 — 125"-1(). C. — 124"-9. For a geometrical explanation of 

 these terms the reader is referred to Proctor's " Moon." or to 

 " Knowledge " for March. 1903. For the mean parallax 

 they use 3422" -70, agreeing well with Newcomb's gravitational 

 value 3422" -68 ; H.ansen's value was 3422" -24 : for the mean 

 semi-diameter they use 933"- 60. which is half a second less than 

 Hansen, and exactly agrees with Cowell's value deduced from 

 fifty years of Greenwich observations; Hansen's value is 

 evidently too great : even the new one is one second greater 

 than the value obtained from occultations, the difference 

 being due to irradiation, which causes all bright bodies to be 

 measured a little too large. On the whole the new tables are 

 a great improvement on Hansen's, and the only doubt about 

 their utility is due to the fact that Brown's tables are nearly 

 ready, and will probably be still more e.xact than these. How- 

 ever, there are some advantages in having two reliable tables ; 

 .sometimes small errors are picked up by comparing them ; 

 thus a small error in Newcomb's Mars tables was found by 

 comparison with Leverrier's tables, which are still used by the 

 Connaissance des Temps; I presume that it will use the new 

 lunar tables, starting « ith the year 1914. 



ERRATA IN LAST MONTH'S NOTES.— Two points in 

 last month's notes need correction. The diagram showing the 

 change of period of Encke's comet was printed upside down. 

 It is now repeated correctly. Secondly, in the list of dates 

 of observations of the satellites of Mars, one accidentally 

 dropped out. The list is therefore repeated ; the dates for 



+ ll"-5cos [l°-37 (t- 

 + 3"-3 cos [5°-6 (t- 



-1790-5)1. Period 263 years. 

 -1856-51]. Period 64 • 3 years. 



They make an interesting suggestion as to the 

 possible origin of these terms: "Consider a swarm 



FiGUKH 1. l-^ncke's Comet. 



