Page:Encyclopædia Britannica, Ninth Edition, v. 16.djvu/832

Rh 802 MOON starting with a provisional approximate solution (that of Delaunay being accepted for the purpose), and substituting the expressions for the moon s coordinates in the funda mental differential equations of the moon s motion as dis turbed by the sun. If the theory were perfect, the two sides of each equation would come out equal. As they do not come out exactly equal, Sir George puts the problem in the form : What corrections must be applied to the expressions for the coordinates that the two sides may be made equal ? He then shows how these corrections may be found by solving a system of equations. The several methods which we have described have for their immediate object the determination of the motion of the moon round the earth under the influence of the combined attractions of the earth and sun. In other words, the question is that of solving the celebrated &quot;problem of three bodies&quot; in the special case when one of the bodies, the sun, has a much greater mass than the other two, and is at a much greater distance from them than they are from each other. All methods lead to a solution of the same general form which we shall now describe. Let us put g the moon s mean anomaly ; g&quot; the mean anomaly of the sun (or earth) ; w the angular distance of the lunar perigee from the moon s node on the ecliptic ; u the angular distance of the sun s perigee from the moon s node on the ecliptic. When no account is taken of the action of the sun the angles g and g 1 increase uniformly with the time, representing in fact the uniform motion of the moon round the earth and of the earth round the sun, while w and w remain constant. When account is taken of the action of the sun all four of the angles change with a uniform progressive motion. In conse quence, the mean orbit of the moon round the earth becomes a moving ellipse whose major axis makes a revolution round the earth in about nine years, and the line of whose nodes makes a revolution in about eighteen and a half years. All the other ele ments of this ellipse namely, its major axis, its eccentricity, and its inclination to the ecliptic remain absolutely constant however long the motion may continue, unless some other disturbing force than that of the sun comes into play. But in the actual motion of the moon there are periodic deviations from this ellipse, which may be represented by an infinite trigonometric series, each term of which is of the form c (sin or cos) (ig+i g 1 +jw+fu), in which the quantities c are absolutely constant coefficients, and positive, negative, or zero. The circular function is, a sine in the expression for longitude or latitude, a cosine in the expression for the parallax. Also, j and / must be both even or both odd in the expressions for longitude and parallax, but the one even and the other odd in the case of the latitude. For example, if we sup pose j, j, and i all zero, we shall have terms of the form c x sin g 1 + c 2 sin 2# + c 3 sin Bg +, &c. To write other terms, suppose i=l, then we have terms of the form &amp;lt;T! sin (g - g) + e a sin (g + g } + e s sin (g + 2g ) +, &c. Taking the case when j = 2 and / = - 2, we shall have terms of the form M! sin (g - g + 2w - 2w ) + m. 2 sin (g - 1g + 2w - 2o&amp;gt; ) + , &c. AH the indices i, i , j, and/ become larger, the coefficients c, e, m, kc., become smaller; but the number of terms included in the theories of Hanson and Delaunay amount to several hundreds. In tho analytical theories, like that of Delaunay, each of the coeffi cients c, e, in, &c., is a complicated infinite series, but in the numerical theories it is a constant number. And the principal problem of the modern theory of three bodies is to find the appropriate co efficient for each of these hundreds of terms. Diction of the Planets on the Moon. For nearly two centuries it has been known from observations that the mean motion of the moon round the earth is not absolutely constant, as it ought to he were there no disturbing body but the sun. The general fact that the motion has been accelerated since the time of Ptolemy was first pointed out by Halley, and the amount of the acceleration was found by Dunthorne. After vain efforts by the greatest mathe maticians of the last century to find a physical cause for the acceleration, Laplace was successful in tracing it to the secular diminution of the eccentricity of the earth s orbit, produced by the action of the planets, lie computed its amount to be 10 per century that is, if the place of the moon were calculated forward on its mean motion at the beginning of any century, it would at the end of the century bo 10&quot; in advance of its computed place. This theoretical result of Laplace agreed so closely with the acceleration found by Lalande from the records of ancient and mediaeval eclipses that it was not questioned for nearly a century. In 181)2 Mr John C. Adams showed that Laplace had failed to take account of a series of terms, the effect of which was to reduce the acceleration to 6&quot; or less. The result was inconsistent with. the accounts of ancient eclipses of the sun, and a cause for the discrepancy had to be sought for. A probable cause was pointed out, first by Ferrel, and afterwards by Delaunay. The former,. in papers published in Gould s Astronomical Journal, and in the Proceedings of the American Academy of Arts and Sciences, showed that the action of the moon on the tidal waves of the ocean would have the effect of increasing the time of the earth s axial rotation or the length of the day, which is necessarily taken as the unit of time. Since, as the days became longer, the moon would move farther in one day, though its absolute motion should remain unchanged, and hence an apparent acceleration would be the result. That this cause really acts there can be no doubt. But the data for determining its exact amount are discrepant. If we take only such data as are purely astronomical namely, the eclipses recorded by Ptolemy between 720 B.C. and 150 A.D., and those observed by the Arabians between 800 and 1000 A.n. the apparent excess of the observed acceleration to be accounted for by the tidal retarda tion amounts to only 2&quot; per century, and may be even less. But this small acceleration is entirely incompatible with conclusions drawn from certain supposed accounts of total eclipses of the sun, notably the eclipse associated with the name of Thales. This is the famous eclipse supposed to be alluded to by Herodotus when he describes a battle as stopped by a sudden advent of darkness, which had been predicted by Thales. If the true value of the co efficient resulting from the combined effect of tidal retardation of the earth and secular acceleration of the moon is less than 10&quot;, then not only could the path of totality not have passed over the field of battle but the greatest eclipse could not have occurred till after sunset. In fact, to represent this and other supposed eclipses of the sun, the acceleration must be increased to about 12&quot;, which is near the value found by Hansen from theory, and adopted in his tables of the moon. But his theoretical computation is un doubtedly incorrect, because in computing in what manner the eccentricity of the earth s orbit enters into the moon s motion he took account only of the first approximation, as Laplace had done. The following is a summary of the present state of the question : The theoretical value of the acceleration, assuming the day to be constant, is, according to Delaunay ......... 6&quot; 17& Hansen s value, in his Tables de la Lune, is ............... 12 18 Hansen s revised but still theoretically erroneous result is 12 56 The value which best represents the supposed eclipses (1) of Thales, (2) at Larissa, (3) at Stikkelstad, is about 117 The result from purely astronomical observations is ...... 8 3 The result from Arabian and modern observations alone is about ............................................................ 7 Inequalities of Long Period. Combined with the question of the secular acceleration is another which is still entirely unsettled namely, that of inequalities of long period in the mean motion of the moon round the earth. Laplace first showed that modern observations of the moon indicated that its mean motion was really less during the second half of the 18th century than during the first half, and hence inferred the existence of an inequality having a period of more than a century. All efforts to find a satisfactory explanation were, however, so unavailing that Poisson, in 1835, disputed the reality of the inequality. But Airy, from his discussion of the Greenwich observations between 1750 and 1830, conclusively proved its existence. About the same time Hansen announced that he had found from theory two terms of long period arising from the action of Venus which fxilly corre sponded to the inequalities indicated by the observations. These terms, as employed in his Tables de la Lune, are 15&quot; &quot;34 sin ( -g- 16&amp;lt;/ + 180&quot; + 3 + 21&quot;-47 sin (8/- ISg + 4 44 ), 36 ) in which g, g, and g&quot; represent the mean anomalies of the moon, the earth, and Venus respectively. During the first few years after the publication of Hansen s tables they represented observa tions so well that their entire correctness was generally taken for granted. But doubt soon began to be thrown upon the inequalities of long period just mentioned. Indeed, Hansen himself admitted that the second and larger term was partly empiri cal, being taken so as to satisfy observations between 1750 and 1850. Delaunay re-computed both terms, and found for the first term a result substantially identical with that of Hansen. But lie found for the second or empirical one a coefficient of only 0&quot; 27, which would be quite insensible. With this smaller coefficient the obser vations from 1750 could not be satisfied, so that, so far as observa tions could go in deciding a purely mathematical question, the evidence was in favour of Hansen s result. But 011 comparing Hansen s tables with observations between 1650 and 1750 it was found that the supposed agreement with observation was entirely illusory. Moreover, since 1865 the moon has been steadily falling behind the tabular place. These inequalities of long period have not yet been satisfactorily explained. The most plausible supixxsi- tion is that they are due to the action of one or more of the larger planets. But the problem of the action of the planets on the mcon
 * * j&amp;gt; and/ are integers which may take all combinations of values