NOVKM'JKK 



I 905 J 



N-A TURE 



81 



exhaustive search I only brought to light one fresh 

 inequality that runs through the errors, and that is 

 to all appearance due to an error in the adopted 

 parallax of the moon. My analysis, however, enables 

 me to say that the solution of the problem of three 

 bodies, as recently completed by E. W. Brown, is 

 final. This might fairly be inferred from its agree- 

 ment with Hansen and Delaunay, and from the 

 numerous equations of verification employed through- 

 out by Brown. But on my analysis a further remark 

 may be based; not only are Brown's expressions a 

 e. inert solution of his differential equations, but those 

 differential equations do really represent, with all 

 necessary accuracy, the problem of three bodies as 

 presented by nature. The problem has been solved. 

 If in the future a method as much superior to Hill's 

 as Hill's is to Hansen's were to be invented, it would 

 no doubt be worked out numerically, but no matter 

 how ingenious it might be, the test of its accuracy 

 would be — does it agree with Brown? 



Another inference may be drawn from what I may 

 call mv empirical lunar theory. As the coefficients of 

 solar terms are verified by Brown's calculations with 

 a probable error of about o"-o4, that is presumably a 

 measure of the accuracy of the constants. Moreover, 

 on comparing the planetary and figure of earth terms 

 with theorv, larger discordances are found, especialh 

 in the fitrure of earth terms and in the Jupiter evection 

 term. There is no special difficulty in obtaining these 

 terms from observation ; they are presumably deter- 

 mined as accurately as the others. Consequently, 

 appreciable errors still exist in the theoretical values 

 of the figure of earth terms and the Jupiter evection 

 term. 



Two suppositions of Hansen's on which he founded 

 alterations of his tables have also been disproved, a 

 mechanical ellipticity of the moon and an eccentricity 

 in the face that it exhibits to the earth. 



I come now to another class of investigations. The 

 theory of the moon is deficient in that it does not 

 explain the cause of a term of period of about 300 

 \r.irs and coefficient 15" which observation shows 

 to exist. This deficiency of theory is an inconvenience 

 in many ways. It renders the determination of the 

 secular acceleration of the moon, and the resulting 

 measurement of tidal retardation, impossible from 

 modern observations. It will be years, possibly two 

 centuries, before from observation alone a really 

 accurate estimate of the missing term can be given, 

 unless, as is much to be hoped, theory accounts for 

 it in the meanwhile. This unknown term renders 

 difficult also the determination of the motion of the 

 node and perigee. The position of the perigee is 

 found bv measuring an arc equal to the mean anomaly 

 back from the mean position of the moon, and it is 

 fairly clear that the unknown term is also an in- 

 equality of the anomaly. Hence the motion of the 

 anomaly contains a periodic part that it is difficult 

 to allow for accurately. I have determined the motion 

 of the node and perigee over a period of 150 years, 

 and I get small differences from the theoretical values 

 recently published by Brown. Possibly the cause that 

 produces the term of long period also produces a 

 small motion of the node and perigee. Hansen 

 assumed an empirical term of 240 years' period for 

 this unknown term, but before Hansen's tables had 

 been in the Nautical Almanac for twenty years, New- 

 comb found it necessary to change the period assumed 

 to 273 years. Each assumption was associated with 

 an argument in the hope that it would turn out to be 

 the correct argument, but both in turn have been 

 disproved. My own idea as to the term is that its 

 period is more nearly 350 years, and I have no sugges- 

 tions to make as to its argument. There are also 



no. 1882, vol. 7,;] 



smaller terms of 40 and 70 years' period approxi- 

 mately, or possibly the errors assume a more compli- 

 cated form still. "The periods are so long that the 

 uncertainty is gre.il. 



The last section of my investigations deals with 

 the ancient solar eclipses and the value of the secular 

 accelerations. The three angles mentioned at the 

 outset of this paper as requiring measurement contain 

 terms proportional to the square of the time. It is 

 evident that these terms become of considerable 

 importance at remote epochs. Also on their accurate 

 determination depend (1) the degree of assistance that 

 astronomy can extend to chronologists ; (2) a numerical 

 estimate of the tidal retardation of the earth's diurnal 

 rotation. 



I have succeeded in showing that the alteration of 

 two of the secular terms renders total, or at any rate 

 central, five ancient eclipses which are partial accord- 

 ing to the existing tables. This may, of course, be 

 .111 extraordinary coincidence, but it seems more 

 natural to suppose that records of the eclipses have 

 come down to us because they really were striking 

 phenomena worth recording— in one case the account 

 says " fire in the midst of heaven," which seems to 

 indicate the corona, and therefore totality. There is 

 also the further fact in favour of these corrections 

 that one of them is confirmed and the other supported 

 by the ancient lunar eclipses. It may be of interest 

 li. mention that the most ancient eclipse of the five 

 was communicated to me from the British Museum 

 after I had deduced corrections from the other four, 

 and that the corrections already found satisfied^ the 

 condition of totality for the newly discovered eclipse. 

 To such an extraordinary piece of luck the words of 

 Virgil seem applicable : — 



" Turne, quod optanti divom promittere nemo 

 Auderet, volvenda dies en attulit ultro." 

 It had occurred to me to wonder whether it was 

 worth while to write to the British Museum, but the 

 chance seemed so small that I was letting the days 

 slip by without doing so. 



Ancient eclipses, " therefore, give an accurate 

 measure of the relative distances of three points, the 

 positions of the node, the sun, and the moon. The 

 next question is, " Where is the equinox relatively 

 to these three points? " My first interpretation of 

 my results proceeded thus :— The position of the sun 

 relatively to the equinox has never been called in 

 question. We may be assumed to know it. There- 

 fore my calculations determine the distance of the 

 node from the equinox. This view of the matter, I 

 now am glad to say, was found on examination to 

 be untenable. In the words of Dante, what I spun 

 in October did not last until the middle of November 

 (the date of the first meeting of the Royal Astro- 

 nomical Society) : — 



" a mezzo novembre 

 non giunge quel che tu d'ottobre fili." 



Purg., vi., 143. 



The position of the node, in fact, may be inferred 

 with certainty from the gravitational calculations of 

 Prof. Brown.' Hence m\ eclipse results determine the 

 position of the sun as well as of the moon. The con- 

 clusion is that the sun's motion is being accelerated. 

 The most obvious hypothesis to account for this 

 observed fact — it does not follow that it is the only 

 hypothesis — is that the aether has a sensible retard- 

 ing effect. It may seem curious that the resistance 

 of" the aether should accelerate the earth's orbital 

 motion, but that undoubtedly would be the effect. 

 The total energy must be diminished, and this implies 

 that the planet falls in towards the sun and conse- 

 quently revolves faster in its orbit. P. H. Cowell. 



