r. - - - - r ~ - - -,n^t^jCSt_n^_ni_ri_ii^n_ji_n_jv^^ REESE LIBRARY OF THE UNIVERSITY OF CALIFORNIA. ,190 . Accession No. 91 021 Class No. LECTUBES ON THE LUNAE THEOEY, HonUon: C. J. CLAY AND SONS, CAMBRIDGE UNIVERSITY PRESS WAREHOUSE, AVE MARIA LANE. 50, WELLINGTON STREET. F. A. BROCKHAUS. THE MACMILLAN COMPANY. Bombay: E. SEYMOUR HALE. LECTURES ON THE LUNAR THEORY BY JOHN COUCH ADAMS, M.A., F.B.S. LATE LOWNDEAN PROFESSOR OF ASTRONOMY AND GEOMETRY IN THE UNIVERSITY OF CAMBRIDGE. EDITED BY R. A. SAMPSON, M.A. PROFESSOR OF MATHEMATICS IN THE UNIVERSITY OF DURHAM. CAMBRIDGE : AT THE UNIVERSITY PRESS. 1900 [All Rights reserved.] Cambridge : PRINTED BY J. AND C. F. CLAY, AT THE UNIVERSITY PRESS. PREFACE. following lectures were collected from manuscripts left -*- by the late Professor J. C. Adams, and are now reprinted without change from his Collected Scientific Papers, Vol. n., pp. i_84. It was thought that the wide interest attaching to the lunar problem reached many besides the professed astronomer, and would justify a separate publication of this short work. It is known that Adams contemplated the publication of some such essay himself, and it must be a matter of regret to all that he never did so. No pains have been spared to present the material properly, but it is unavoidable that it should appear from the hands of an editor in a less perfect form than if the author had issued it himself. Yet, allowing for this disadvantage, I think those best qualified to judge will consider this work fully worthy of Adams's great name. Of current elementary theories it may be said that they leave off where the difficulties of the subject begin, that is to say, where the various cases of slow con- vergence have been exposed, but not dealt with. It is perhaps not too much to say that these lectures carry us to the point where such difficulties end, in an adequate evaluation of all the chief constants. They leave the problem effectively solved and not merely stated, and shew the path clear for the for- mation of a detailed theory, if that is desired. R. A. SAMPSON. DURHAM. 8 October, 1900. 91021 CONTENTS. LECTURE PAGE I. HISTORICAL SKETCH 3 II. ACCELERATIONS OF THE MOON RELATIVE TO THE EARTH. 7 III. THE SUN'S COORDINATES IN TERMS OF THE TIME . . 13 IV. THE VARIATION 16 V. THE VARIATION (continued} 20 VI. THE VARIATION (continued) 25 VII. CORRECTION OF APPROXIMATE SOLUTIONS .... 30 VIII. THE PARALLACTIC INEQUALITY 34 IX. THE PARALLACTIC INEQUALITY (continued} ... 39 X. THE ANNUAL EQUATION 43 XI. THE EQUATION OF THE CENTRE AND THE EVECTION . 48 XII. THE EVECTION AND THE MOTION OF THE APSE . . 54 XIII. THE MOTION OF THE APSE, AND THE CHANGE OF THE ECCENTRICITY 61 XIV. THE LATITUDE AND THE MOTION OF THE NODE . . 68 XV. MOTION IN AN ORBIT OF ANY INCLINATION ... 72 XVI. MOTION IN AN ORBIT OF ANY INCLINATION (continued] . 76 XVII. ON HILL'S METHOD OF TREATING THE LUNAR THEORY . 81 XVIII. ON HILL'S METHOD OF TREATING THE LUNAR THEORY (c,ont.} 85 - LECTURES ON THE LUNAR THEORY. [LECTURES on the Lunar Theory were given by Adams from 1860 with few intermissions until 1889. Originally their aim was to illustrate geometrically the analytical processes and thereby render them more comprehensible, and they included some elegant theorems on the geometry of conies which have since become common property ; but every year several lectures were rewritten, and thus the whole fabric gradually changed into the form in which it is here presented, the form, prac- tically, in which he gave them last. Perhaps it is superfluous to say that these Lectures stand upon a different footing to treatises that are intended to form the basis of Tables. With such, completeness is the first object and manner of presentation is secondary. Immense as is the labour of forming a treatise of this description, there exist several that leave little to desire in respect to fulness of detail. Indeed it may be suspected that their very perfection in the quality they profess has stifled to some degree the proper development of the subject, because at first sight it suggests that there is little left to do in the Lunar Theory, unless one is prepared to track down the inconsiderable errors that have eluded his Masters. This seems a mistake; the methods most suitable for the whole task adapt themselves comparatively ill to each detail of it, and there seems much that remains to be done in respect to inventing methods suitable for attacking separately, as far as they permit of separate attack, the many difficulties into which the theory divides at the outset, and thence perhaps approximating to A. L. 1 2 LECTURES ON THE LUNAR THEORY. a more adequate knowledge than we now possess of the relative motion of Three Bodies. So far, with the notable exception of Dr G. W. Hill and those that have followed him, we have seen comparatively little effort in this direction. This was the cardinal feature of Adams's plan, and his lectures shew the methods he had gradually elaborated to accomplish it. They separate the inequalities from one another as far as possible, and are content with indicating the manner in which these separate inequalities afterwards combine. To shew that, with so slight an apparatus and within so small a compass, the result is no mere sketch, we need but set side by side the coefficients of longitude found in these Lectures and the corresponding terms in Delaunay's Theorie. Adams. Delaunay. Variation, coeff. of sin 2D sin4D Parallactic inequality, sin D sin 3D sin5D Annual equation, sin I' Evection, Further, Motion of Apse, Motion of Node, 2106-4 2106-25 8-74 8-75 -124-90* -127-62 0-73 0-84 o-oi o-oi -658-9 - 659-23 152-09 152-11 - 21-57 - 21-63 4596-6 4607-77 175-1 174;87 1-c 008554 003997 008572 003999 For those to whom the difficulties of the Lunar Theory are known, these numbers need no comment. No Manuscript exists of Lecture I. It is taken substantially from my own notes of 1889.] * With Delaunay's value of the Sun's Parallax, viz. 8"-75, LECTURE I. HISTORICAL SKETCH. [THE Lunar Theory may be said to have had its commence- ment with Newton. Many irregularities in the Moon's motion were known before his time, but it was he that first explained the cause of those irregularities and calculated their amounts from theory. Of the inequalities which are due to the action of the Sun, the first, which is called the Evection, was discovered by Ptolemy, who lived at Alexandria in the first half of the second century of our era, under the reigns of Hadrian and Antoninus Pius. At a very early period the relative distance of the Moon at different times could be told from the angle it subtended, and its orbit could thus be mapped out. By such means Ptolemy found that its form was not the same from month to month, and that the longer axis moved con- tinually though not uniformly in one direction. He represented this change by a motion of the centre of the ellipse, as we would put it, in an epicycle round the focus, obtaining thus a variable motion for the longer axis and a variable eccentricity. The representation of position by means of epicycles is intimately related to the modern method of developing the coordinates in harmonic series ; thus if we have x A l cos (nj, + a^ + A 2 cos (n 2 t + 2 ) + . . . y = A 1 sin (nj + a x ) + A z sin (nj + 02) + ... the motion of the point (x, y) is that on a circle of radius A l with angular velocity n lt around a centre which moves on a 12 4 LECTURES ON THE LUNAR THEORY. [LECT. circle of radius A 2 with angular velocity n 2 , and so on; and if, more generally, we have x = A l cos (njt + !)+... y = B l sin (n^t + a 1 )+ ... we may reduce this case to the former by rewriting a? = (4i + B 1 ) cos (nj + a 1 ) + ( y = ^ (A + #1) sin (w^ + aj + g (4 x - x ) sin (- ^ - i) + . . . . Probably we have here the reason why circular motions and epicycles were first employed. Tycho Brahe (1546 1601) discovered the existence of another inequality in the Moon's Longitude quite different from the Elliptic Inequality and the Evection. He found it bore reference to the position of the Sun with regard to the Moon ; so that when the Sun and the Moon were in conjunction or opposition or quadratures the position of the Moon was quite well represented by the existing theory, but from con- junction to the quadrature following, her position was more advanced than the place assigned to it, reaching a maximum of some 35' about half-way; and in the second quadrant it was just as much behind. This inequality he called the Variation; it was the first that Newton accounted for theo- retically, and if we were to suppose the Moon and Sun to move, except for mutual disturbance, in pure circles in the same plane, it is the only one that would present itself. The next significant step was made by Horrox (1619 1641) who represented the Evection geometrically by motion in a variable ellipse, and gave very approximately the law of varia- tion of the eccentricity and the motion of the apse. He supposed the focus of the orbit to move in an epicycle about its mean place. Newton's Principia did not profess to be and was not intended for a complete exposition of the Lunar Theory. It was fragmentary; its object was to shew that the more I.] HISTORICAL SKETCH. 5 prominent irregularities admitted of explanation on his newly discovered theory of universal gravitation. He explained the Variation completely, and traced its effects in Radius Vector as well as in Longitude; and he also saw clearly that the change of eccentricity and motion of the apse that constitute the Evection could be explained on his principles, but he did not give the investigation in the Principia, even to the extent to which he had actually carried it. The approximations are more difficult in this case than in that of the Variation, and require to be carried further in order to furnish results of the same accuracy as had already been obtained by Horrox from observation. He was more successful in dealing with the motion of the node and the law of change of inclination. He shewed that when Sun and Node were in conjunction, then for nearly a month the Moon moved in a plane very approximately, and that the inclination of the orbit then reached its maximum, namely, 5 17' about; but as the Sun moved away from the Node the latter also began to move, attaining its greatest rate when the separation was a quadrant, and that at this instant the inclination was 5 very nearly. He also assigned the law for intermediate positions. The fact that there was no motion when the Sun was at the Node, that is, in the plane of the Moon's orbit, confirmed his theory that these inequalities were due to the Sun's action. When we spoke of Newton's results as fragmentary and incomplete, let it not be understood that he gave only very rude approximations to the truth. His results are far more accurate than those arrived at in elementary works of the present day upon the subject. After Newton, Clairaut (1713 1765) treated the Lunar Theory analytically. He readily found the Variation and many other inequalities, but met with a difficulty in determining the motion of the apse. At first he made its mean motion only about one-half of the observed value, and supposed that this indicated a failure of Newton's law of the inverse square of the distance ; but soon he recognized an error, caused by omission of terms which he had imagined would not affect 6 LECTURES ON THE LUNAR THEORY. [LECT. I. the result. When these were included the calculated amount was nearly doubled. The first Tables of the Moon which were sufficiently accu- rate for use in determining longitudes at sea by observation of Lunar Distances were those of Mayer. They obtained a prize offered by our Board of Longitude, and were published in 1770 by Maskelyne, the Astronomer Royal. The first Theories which gave the Moon's place with an accuracy equal to that of observation were those of Damoiseau and Plana. The former was published in 1827, preceded in 1824 by Tables; the latter was published in 1832. Hansen's Tables, which are those now used, were constructed from theory and were published in 1857 at the expense of the British Government.] LECTURE II. ACCELERATIONS OF THE MOON RELATIVE TO THE EARTH. WHEN three bodies move under their mutual attraction, their motions are unknown to us except in the cases when they are approximately elliptical ; but this restriction includes almost all the most important cases in the Solar System. If one body of the system is greatly predominant and if the lesser bodies are not close together, the centre of gravity of the greater body may be taken as a common focus around which the others move in approximate ellipses. Or again, if two bodies lie close together, their relative motion may be approximately the same as though they were isolated, although the system contains a third greatly predominant body; for their relative motion is affected by the difference of the attrac- tions of the central body upon them and not by the absolute value of those attractions. The Sun and Planets are an example of the first kind; the Earth, Moon and Sun of the second. The Earth and Moon describe orbits round the Sun which are approximately ellipses, and the Moon might be regarded as one of the planets; but this point of view would not be a simple one ; the disturb- ing action of the Earth would be too great, though it is never so great as the direct attraction of the Sun, that is to say, never great enough to make the Moon's path convex to the Sun. The more convenient method is to refer the motion of the Moon to the Earth, and counting only the difference of 8 LECTURES ON THE LUNAR THEORY. [LECT. the attractions of the Sun upon the Earth and upon the Moon, to find how this distorts the otherwise elliptical relative orbit. This is the method of the Lunar Theory. The position of the Sun must be referred to the same origin ; but since the Earth describes an ellipse about the Sun which is disturbed by the action of the Moon, if we choose as origin the Earth's centre, we must allow for the disturbance of the Sun's position by the Moon. This correction may be evaded by choosing as origin, not the Earth's centre, but the centre of gravity of the Earth and Moon, with respect to which the Sun E describes a curve so closely el- liptical that no allowance is required. For, if $, E, M denote respectively the Sun, Earth, and Moon, and G the centre of gravity of E and M, the accelerating forces of S are on E S/SE* in ES, and these imply accelerations of G of amount M S now the accelerations of S are E/SE* in SE, M/SM* in SM; hence the acceleration of G relative to S is S+E+M E * parallel to S S+E + M M S + E + M /_, OE , GM\ . ., E + M ( E 'SE*- M 'SM*) IU - GM > . m GS ' SG\ II.] ACCELERATIONS OF THE MOON RELATIVE TO THE EAllTH. 9 Let EM = r, SG = r', SGM = co ; then M Hence i i r. E + _vV 3 . 15 i == M 1 "ra? 8cosa) M ]- r\V 3 15 \ 1 /)l-2 + T cos ^J + ...... J ; and the accelerations of G are ...... ]<* ...... ]- Now r// is approximately jr ; neglecting the square of this quantity, we see that S moves about G in a pure ellipse. Consider now the accelerations of the Moon relative to the Earth ; subtracting the accelerations of the Earth from those of the Moon, we find MO EG to let E+M = /j,, S = m f ; then these become parallel to OS. 10 LECTURES ON THE LUNAR THEORY. fLECT. In the accompanying spherical triangle, let G be the centre of the sphere, SM f the ecliptic, and M' the projection of M. Let I/u be the projection of ME on the plane of the ecliptic; the longitude of the Moon as seen from the Earth, & the longitude of the Sun as seen from G, s the tangent of the Moon's latitude MM'. Then r = (l+ s 2 )^ u~\ cos to = cos (0 - 0') (1 + a*)-*, and the accelerations of M relative to E are (- 1 (1 + ") + ~ cos 2 ((9 - 0')) + ...... 1 parallel to OS. Call these quantities U and F respectively; then if we resolve parallel to M'G, perpendicular to M'G in the plane of the ecliptic, and perpendicular to the plane of the ecliptic, we have the following quantities which we call P, T, S ; viz. : P= Z7(l+*)-*-F cos (0-00, T=- Fsm(<9-0'X and also S-Ps = Vs cos (0-0'). From these we find 3 cos (0- 0') + cos 3(0 -0')} + ...1 , j II.] ACCELERATIONS OF THE MOON RELATIVE TO THE EARTH. 11 Hence with the time as independent variable we have the equations of motion rl i = P d? \dtj r dt V 'dt Or we may write these with as independent variable ; let 7/1 so that T whence again, whence ds 12 LECTURES ON THE LUNAR THEORY. [LECT. II. or the equations of motion may be written H dH_T ~ 3> d0' Our problem is to discuss these equations and to obtain from them expressions for the Moon's position at any time. The integration is best effected by observing what kinds of terms will disappear on substitution in the equations, and then assuming for the desired expressions for the coordinates a series of such terms multiplied by undetermined coefficients. Our procedure will be to discuss one by one the irregularities which can be isolated from one another. This will permit a survey of the entire field without involving needless complexity ; but if the Lunar Theory is to be accurate, the combinations of such terms with one another must also be included, and the number of terms employed and the labour of manipulating them becomes very great. LECTURE III. THE SUNS COORDINATES IN TERMS OF THE TIME. To obtain the Moon's coordinates in terms of the time from the equations found in Lecture II., we must substitute in the expressions for the forces the developments of the Sun's co- ordinates which we now proceed to give. Employing as coordinates r', 0', of the last lecture, we have seen that the Sun's motion may be regarded as purely elliptical, so that a _ 1 + e f cos (6' CT') /" 1-e' 2 & - w ' = n f t -*r' + 2e' sin (n't -') + z e> * sin 2 ( n/t ~ OT/ ) + rr in which we have written for convenience n't in place of n't + e. The quantities that enter the equations are sin Making the substitutions we find without difficulty ?' cos (n't -&'+-- e' z cos 14 LECTURES ON THE LUNAR THEORY. [LECT. Y cos 2(0-r)=(i-L'A cos 2(0-vo r ) sin v ' \ 2 / sin v sm I r*OQ - g a' {2 (*-'*) + ('-' ' {((9 -n't)- (n't-*/ sm (V ~o~ v 8 sm 11 , a cos 8 sin r / sin sm 1 e"> S {3 (0 - n't) + 2 (n't - w')) o sm These quantities are to be substituted where they occur in the expressions for the forces found in Lecture II. Let us now make a few general remarks upon the result of the substitution. III.] THE SUN'S COORDINATES IN TERMS OF THE TIME. 15 It will be observed that the disturbing forces all involve the coefficient m'a'~ 3 . It is very important to notice that the Sun's parallax is not required for the evaluation of this quantity. By Kepler's Third Law it is derivable from observations of the Sun's mean motion alone. Other terms however, namely those with the coefficient m'/a'*u, involve the Sun's parallax directly ; and that constant may be obtained by comparing the observed with the theoretical values of the coefficients of those in- equalities, with an accuracy probably greater than that of any other method. The mean disturbing force is radial, and is equal to 4- - //8 f 2 6 or the mean effect of the Sun's disturbance is to diminish the Moon's gravity towards the Earth ; and to diminish it more, the greater is the eccentricity of the Sun's orbit. Now e' has been diminishing for ages; hence the Moon's gravity towards the Earth has been increasing, and its average time for accom- plishing a revolution about the Earth has been diminishing. This is one cause of the Secular Acceleration of the Moon's mean motion which Halley derived from the records of ancient eclipses. It may also be noticed that the coefficient of the chief r* periodic part of the disturbing force, which involves 1 5 e' 2 , 2* increases as e diminishes. Finally let it be observed that the term with argument 2 (0 - n't) + 2 (n't - w'), which does not involve the Sun's Mean Longitude, is absent from the development of ( ~ ) C S 2 (d - &). LECTURE IV. THE VARIATION. THE Variation is the first inequality we shall consider ; this is the inequality which is independent of eccentricities and mutual inclination in the orbits of the Sun and Moon. Let us first take the equations in the first form in which they are given in Lecture II., namely with t as independent variable : dt* dt rdt\ dt, we omit the equation of motion in latitude, and in the ex- pressions for P, T we suppose 5 = 0; moreover it is possible and convenient to discuss separately the terms that involve the Sun's parallax; let these be omitted and we have and if e' = 0, T' = a, m'/a s = n'\ & = n't + e , d*0 ZdrdO 3 3io+-j7^ji = - rf^ 2 r dt dt 2 these are the equations to discuss. LECT. IV.] THE VARIATION. 17 Assume as a first approximation 6 = nt + e + 6 2 sin [2 (nt + e) - 2 (>' + e')} = nt + e + 6 2 sin 2i/r, say ; J 5- 1 [l+o, 008 2*1 and we shall suppose a 2 , 6 a so small that in the first instance we may neglect their squares and products. Substitute in the equations ; then 4 (n n'Y a 2 cos 2i/r [ri* + 4>n(n ri) 6 2 cos w + ^ 2 cos a ft 4 ( w _ n 'y j 2 s i n 2^ 4- 4 (n - w') wa 2 sin 2i/r = - - ^ /2 sin 2- Hence, equating the coefficients of similar terms, we have which gives the relation between n the Moon's mean motion, and - , the mean of the reciprocal of the distance ; also CJb (n-ri)b,= |^' 2 ....... (1), n-n')a 2 = -|n' 2 ....... (2). From (2) 4>n (n ri) b 2 kn z a*i = = - , . Add to (1), and substitute for //./a 3 ; A. L. 18 LECTURES ON THE LUNAR THEORY. [LECT. n 3 tt' 2 n- 3 , 2 n(2n-ri} _ 1 _ .3 n 2 rt ^ /.. ._/\o ^^"^^^^^^"^^77"^ T +H^ Calling = m, we have 3 2-m = -^ m . 2 1-m* Q 11 2 3 - 8m + -7T- m 2 3 2-m 1 3m h r + COS and from the second equation -4(n-') t & a sin2^ , r r . i ) n + 4 (n - n ) \na. 2 sin 2-Jr + )( w') a 2 6 2 - - na 2 2 ^ sin 4i|r I I 2 J J Q -f x n /2 [sin 2i|r + 6 2 sin 4-^r]. 22 LECTURES ON THE LUNAR THEORY. [LECT. The coefficient of cos 2^/r in the first of these expressions, and that of sin 2^/r in the second, are respectively 4(n - nja,- *n (n - ri) 6 2 + 3 ^ a 2 - | n'\ a & g and -40- ft') 2 & 2 + 4w (n - n') a, + ^ rc' 2 , and these are evidently reduced to zero by giving a 2 , b 2 the values previously found, if we substitute for //,/a 3 the approxi- mate value n 2 + ~ ?i' 2 . To find the more correct value of //,/a 3 , 2 equate to zero the constant term in the first expression ; that is (l + | a/) = ti 2 + i ri' 2 - 2 (rc - ^O 2 2 2 + 2 (w - n') 2 6 2 2 - 1 ^' 2 6 2 = " + i w /2 + 2 (n - rc 7 ) 2 Ugmj + m^) a 2 2 - ^ m^ 1 . Hence we see that /JL/O? differs from ti 2 + 5 w /2 only in terms of the fourth order, if we consider Wj a quantity of the first order and consequently a 2 , 6 2 quantities of the second order. Hence also by taking in the multiplier of a 2 , when we equate to zero the coefficient of cos 2>Jr, we only neglected a quantity of the sixth order in m lt and the error in the resulting values of a 2 , 6 2 is of that order. We see that the substitution just made in our equations leaves outstanding terms of the fourth order in cos 4^ and sin 4-^. In order to get rid of these we must add terms of this form to the assumed values of l/r and #, respectively. Suppose that - = - [1 + a. 2 cos 2i/r -f a 4 cos = nt + e + 6 2 sin 2i/r + 6 4 sin V.] THE VARIATION. 23 where, as we shall find, a 4 and 6 4 are small quantities of the fourth order. It may be readily seen that the additional terms introduced are the following : 1 d?r in -j 16(n %') 2 a 4 cos4^, V Cut ?i 4 cos and also that we may neglect the terms added in the ex- pressions for | rc' 2 cos 2 (0 - n't - e), | n'* sin 2 (0 - ^ - e 7 ). L -j If we write the terms thus produced along with the several terms left outstanding in our equations and then equate the whole to zero, we have 1 6 (n - nj a,-8n(n- n') 6 4 + ^ 3a 4 - 6 (n - nj a* - 2 (n - nj bj a from which we must determine a 4 and 6 4 . Put = M2 + n ' a ' 24 LECTURES ON THE LUNAR THEORY. [LECT. V. and divide both equations by (n n') 2 ; we get I 16 + 3(l + 271^ + 1^ 1 a 4 -8(l + w,)&4- 2 - 3(1 Cl \ 9 1 +2 m i) a 2+jg^i 4 =0. Simplify and multiply the last equation by = (1 -|- mi), 19 + 6m! + mA a 4 - 8 (1 + m,) 6 4 ~ (-^ + m l - i m 27 -3(1 (4 -f 8m! + 4w! 2 ) a 4 - 8 (1 + m^ 6 4 + (1 + 2m 1 + mf) a 2 2 Subtract the latter from the former and 6 4 will be elimi- nated ; we get . 1 A /15 3 15 - 2j = i 4 - i 1 i which gives a 4 ; and this being known 6 4 is found from 1 1 3 / 1 \ 9 6 4 = - (1 + W 1 )a 4 + Q (1 + mjaf +=-^( 1 +^m l ) m^a 2 + ^r^ mf. 2 X 8 16V 2 / 2o6 Taking m = '07480 as in Lecture IV, we find a 4 = -00004,580, 6 4 = -00004,237 = 8"-740. LECTURE VI. THE VARIATION (continued). LET us consider the problem of the Variation over again, taking now as independent variable. The equations of motion are given in Lecture II : d?u P T du P 1 n'* 3 ri 2 where _ =,,___--_ cos 2 (0 - 0'), T _3nf* u 3 ~ 2 U* so that the second equation may be written Our aim is to express t and u in terms of 6 and constant quantities. Now since the orbit of the Moon does not differ widely from a circle we may write the difference of nt + e from 6, and the difference of au from unity as series of small periodic terms depending upon 6. Inspecting the form of the equations, it is evident that these periodic terms are of argument 2 (0 6') and its multiples ; that is nt + e = 6 + periodic terms of argument 2 (6 0'), &c. ; but n't + e' = f ' 26 LECTURES ON THE LUNAR THEORY. [LECT. therefore - & = (1 - m) 9 - j3 + periodic terms of argument 2 (1 - m) - 20, &c., where we have written j3 = e'- me ; this constant ft is associated with (1 m) wherever the latter occurs ; for brevity in writing, we shall omit it. We may then assume as a first approximation au = 1 4- Og cos (2 2m) 0, nt + e = 6 + 6 2 sin (2 - 2m) 6 ; whence 2 ((9 - 0') = (2 - 2m) - 2m 2 sin (2 - 2m) 0, cos 2 (0 - 0') = mb. 2 + cos (2 - 2m) - mb 2 cos (4 - 4m) 0, sin 2 (0 - (9 X ) = sin (2 - 2m) 6 - mb 2 sin (4 - 4m) 6, n g| = 1 + (2 - 2m) 6 a cos (2 - 2w) ft Substitute in the right-hand member of the second equa- tion : ~ 2 ^P = - 3m 2 [sin (2 - 2m) 4- (2 - 3m) 6 2 sin (4 - 4m) 0]. Therefore H*\ 3m 2 3 2 - 3??i = ~ C S 2 ~ 2 lm 4 a which we may write /H 2 \ log e ( -T- J = 2h 2 cos (2 - 2m) + 2/t 4 cos (4 - 4m) 0, where h is an arbitrary constant of integration, A 2 is a known quantity, and h 4 involves b 2 . If we take as a second approxi- mation au = I+a 2 cos (2 2m) + a 4 cos (4 4m) 0, w$ + e = -H 6 2 sin (2 - 2m) + 6 4 sin (4 - 4m) 0, the above value of log e (H 2 /h?) will not require modification and will supply equations of condition for determining the coefficients a 2 , &a> a*, b 4 . VI.] THE VARIATION. 27 dt n no? h 1 I nii tt il d0~Hu*-~h H(au)*' so that / dt\ /na?\ 1 (H*\ \ ctu / \ ti / *2i \ ii / but log, (n < JH"\ = - (1 - m) 2 6 2 2 + (2 - 2m.) 6 2 cos (2 - 2m) + [(4 - 4m) 6 4 - (1 - m) 2 6 2 2 ] cos (4 - 4m) 0, a. 2 / a 2 \ logg au = -j- + a 2 cos (2 2m) 0+[&t-r] cos (4 4w) 0. 4 \ 4 / Hence we find '-joA (4 - 4m) 6 4 - (1 - m) 2 6 2 2 = - h^ - 2a 4 + ^ a 2 2 . 22 The remaining equations of condition that we require are obtained from the first equation of motion ; this may be written d- (au) Now au 1 + a z cos (2 2m) 4- 4 cos (4 4m) 6, whence ^^ = - (2 - 2m) a 2 sin (2 - 2m) - (4 - 4m) a 4 sin (4 - 4m) 0, ^^ = - (2 - 2m) 2 a, cos (2 - 2m) - (4 - 4m) 2 a, cos (4 - 4m) 0, and L' ^Y = m 2 [1 + (4 - 4m) 6 2 cos (2 - 2m) 0], in 2(0-0')= m 2 [sin (2-2m)0+(2 - 3m)6 2 siii(4- 28 LECTURES ON THE LUNAR THEORY. [LECT. = m 2 [(2 - m) 6 2 + cos (2 - 2m) + (2 - 3m) 6 2 cos (4 - 4m) 0], -^ = ^ [1 + A 2 2 - 2A a cos (2 - 2m) + (h* - 2h 4 ) cos (4 - 4m) 0]. Substitute these in the equation above, and equate the coefficients of corresponding terms, - (2 - 2m) 2 a 2 + (l + * m 2 ) a 2 + | m 2 + (2 - 2m) m 2 6 2 = ^ (- - (4 - 4m) 2 a 4 + l + m ' a 4 + ^ 2 (2 - 3m) 6 2 + If we neglect at first terms of the fourth order, we find from the first of these equations From the earlier set of equations we have (2 - 2m) 6 2 = - h - 2a 2 ; substitute this in the second equation above. We get -(2-2m) 2 + l+|m 2 - 2m 2 ja 2 - or 3 m 2 3 2 2-m wfi 21-m~2 m l-m' = ^ m so that 3 n 2-m and 2 l-m 11 3 - 8m + -- m 2 3m 2 1 - m ' 8 (1 - m) 2 ' VI.] THE VARIATION. 29 These are numerically equal to the quantities denoted by the same symbols in Lecture IV, but 6 2 bears the contrary sign. We further find {(1 + 3m - 2m 2 ) a, + (8 - 15m + 6m 2 ) 6 2 ), T! J. 7?l 1 33m 2 , i ^ + (6 or reduced to numbers a 4 = - -00002,210, 6 4 = -00005,414 = n"-i7. Finally let us exhibit the relation between the constants employed in this investigation and those of Lectures IV, V; to distinguish them, attach accents to the latter, so that = nt + + b 2 ' sin 2^ + &/ sin 4>|r, = 1 + a cos 2i|r + a/ cos 4-\Jr, and, omitting the constant /3 as before, (1 - m) = >|r + (1 - m) V sin 2-^ + (1 - m) &/ sin 4i|r. Then 2i|r = (2 - 2m) - (2 - 2m) &/ sin (2 - 2m) 0, sin 2-^r = sin (2 - 2m) - (1 - m) 6 a ' sin (4 - 4m) 0, cos 2i|r = (1 - HI) 6 2 7 + cos (2 - 2m) - (1 - m) 6 2 r cos (4 - 4m) 0. Substitute in the equation for 6 ; we find nt + = 0-b. 2 ' sin (2 - 2m) 6 - [&/ - (1 - m) 6 2 /2 ] sin (4 - 4m) 0, and similarly a'u = 1 + (1 - in) a 2 'br + a 2 ' cos (2 - 2m) + K - (1 - m) o 2 / 6 2 / ] cos (4 - 4m) 0. We observe that a' differs from a by quantities of the fourth order, LECTURE VII. CORRECTION OF APPROXIMATE SOLUTIONS. WE may simplify the equations we have been dealing with, by a proper choice of units. Let the unit of distance be the radius of the circular orbit which the Moon, if undisturbed, would describe about the Earth in its actual periodic time ; then yLt = 7l 2 . Also choose the unit of time so that n ri = 1, so that, if we take as the result of observation of the mean motions of the Sun and Moon, ri : r& = -07480,13, we get n' = '08084,9 = m u where m l is the quantity so called in Lecture IV; and yu = 1-16823,4. We shall frequently 'adopt these simplifications in what follows. Now let l = log e (r/a), so that i^_^ l^r_^ (dlf P = P. -31. r~dt~di' rd&~dP \dtl % r>~ a* e and the equations discussed in Lecture IV become dt 2 * (dt) ~ \dt) + d?Q g dl dO oft 2 + Jt~di 5r*- I = 0, where co = &'. LECT. VII.] CORRECTION OP APPROXIMATE SOLUTIONS. 31 Now these equations are defective, for they have been formed by omitting certain terms from the complete equations as given in Lecture II. Hence, calling 1 , the values of I, 6, which we have proved in Lecture IV to be solutions of the above equations, if we substitute 1 , in the complete equa- tions of Lecture II, residuals are left, say X and T respectively. And if I, 6 be solutions of the complete equations, and if we write e = + so, where 81, BO are small quantities whose squares and products may be neglected in the first instance, we obtain the following equations for determining SI, SO, the corrections to approximate solutions 1 , # already found : *--. Now let us write , dt r 3 a? where c = (1 + rij + $ n' 2 . 2i The quantity v consists wholly of periodic terms of the form cos 2ity multiplied by small coefficients; w contains, besides periodic terms, a small constant term, which however might be removed if we were to choose c as the constant part of yu,/r 3 in place of according to the definition above. Let S'l, S'O be quantities defined by the equations then S'l, S'd are approximations to the complete corrections SI, SO, which if substituted in the equations that give those cor- rections will leave residuals, say X' and F', where 32 LECTURES ON THE LUNAR THEORY. [LECT. _ at at at . ^n ~ do I .0/0 rt C-//1 ' = 2 -^ -- + 2v -- + 3n 2 cos 2o>8 0. We see that their value is known when 87, 8'# are determined. Now 87, 8'0 may be determined as follows. Let X = p 4- %>i cos ity, . Y 2^ sin ity, where i takes all positive integral values ; and assume 87 = a 4- 2c^ cos t-v/r, 8'0 = S6f sin i^. Then substituting and equating coefficients, the constant term gives p - 3ca = 0, and the terms in ity give Pi - Pa* -2(1+ n 1 ) ibi - ^ca { = 0, qi-i*b,i-2(I+ri)iai =0; the second of these may be written subtract from the first and we have or at = and & < = / We see that a it bi will be of the same order of small quantities as pi, qi, in general. And therefore the coefficients of the terms of X', Y' will be of order higher than those of Z, 7. Proceed then to determine further corrections 8"Z, 8"0 satisfying the equations VII.] CORRECTION OF APPROXIMATE SOLUTIONS. 33 then if 8'l + 8"l, 8'0 + 8"0 are substituted in the complete equations for SI, 86 the residuals become 2 T ~ ^ ~ * w *" 1 + 3 ' 2 sin at at at Y" = 2 a a a expressions which, if developed in series of cosines and sines of multiples of ty, will have coefficients of higher order than the corresponding coefficients in X', Y'. The like process may be repeated until the residuals become insensible; we then have sensibly correct values of 81, 80, giving We may now take into account squares and products of the small quantities 81, 80 by treating 1 Q + 81, + &@ a s given approximate solutions just as we have here treated 1 Q , ; substitute them in the complete equations of motion, and determine the residuals X, Y which they leave. These resi- duals will form the basis of a second approximation, and the operation may be repeated until no further correction is necessary. It is to be observed that if 81, 80 depend upon some such constant as the eccentricity of the Earth's orbit around the Sun, or the parallax of the Sun, then successive approximations yield correctly and separately the terms which depend upon the first, second, ...... powers of that constant. A. L. LECTUHE VIII. THE PAUALLACTIC INEQUALITY. WE shall now apply the method of the last lecture to find the terms in the Moon's coordinates which depend upon the parallax of the Sun. The values of I, found in Lecture IV are 1 Q = \og e (r/a) = - a 2 cos 2-^, = nt -f e 4- 6 2 sin 2>/r, and these satisfy the equations of motion in which the terms involving the Sun's parallax are omitted. Hence the residuals they leave from the complete equations are X = -W 2 - j| cos (0 - 00 + ^cos 3(0 -0')j, a (o o j F = \n* - \\ sin (0 -O f ) + ~ sin 3 (0. - 0')} , a (0 o j E-Ma where X = --, =y . E -H M a' Now from above -0' = ^ + & 2 sin2i/r; hence we have sin (0 - 6') = sin i|r + - 6 2 (sin >/r + sin 3^), COS (0 0') = COS i/r - - b 2 (COS ^ - COS 22 sin 3 (0 6') sin 3>|r -f - b. 2 (sin ty + sin 5i/r), COS 3 (0 0') = COS 3i/r '- b. 2 (COS l/r COS 01/r); VIII.] THE PARALLACTIC INEQUALITY. 35 and 15 9 9 \ /45 , 15 + 16 * - 16 V cos 3 * + lie b - - i6 cos '- I sin (ft - 0') + ^ sin 3 (ft - ^) = (J - | 6 2 - | ,) si sn 15 3 , 3 \ . /45 , 15 \ . + 16 6 * ~ 16 ^ sm 3t + lie 6z + 16 " j sm *- Assume &\ = Xa x cos -^ H- Xa 3 cos 3i|r, 80 = \6j sin i|r + \6 3 sin 3i/r, neglecting for the present the terms in 5-^r. In the present case it happens that it is more advantageous to substitute these expressions directly in the complete equations for SI, BO given in the last lecture than to follow exactly the process for finding them by successive approximation. Omitting the factor X, we get a x cos t/r -f- 9a 3 cos 3-^ -f 4a 2 sin 2^ [a x sin ^ + 3a 3 sin CL CL -[2(1+ ri) + 46 2 cos 2^/r] [&! cos ^ -|- 36 3 cos /2 [sin 2^/r + 6 2 sin 4-^r] [6 a sin ty + 6 3 sin ~~ n ' 2 vTfi ^ 2 ~" Tfi a *i cos ^ = ^' >! sin yjr 9b s sin 3-\/r + 4a 2 sin 2^ [6j cos >|r + 36 3 cos 3>/r] + 3w /2 [ 6 2 + cos 2-^r -i- 6 2 cos 4i/r] [6 X sin >/r + 6 3 sin 3^/r] +[2 (1 + n') + 46 2 cos 2>/r] [^ sin >/r + 3a 3 sin 3-^] , /45 , 15 2 ^- I g 2 32 36 LECTURES ON THE LUNAR THEORY. [LECT. If we equate to zero the coefficients of cos ifr and cos 3-^ in the first, and those of sin i/r and sin 3^/r in the second, we obtain the following equations for a lt bi, a 3 , b 3 ; the terms in o-^r remain outstanding, and the effect of a 5) b 5 in modifying the other co- efficient is neglected. a, [2 (1 + n') - 26 2 ] +a,[66J ^ 3 a,] + a 3 [6 (1 + n')] -6 3 [9 If we require the formal values of a lt b lt a S) 6 3 , we must substitute for a 2 , 6 2 , yit/a 3 the expressions we have found for them, and it will then be best to develope the coefficients in ascending powers of ri. But it is difficult to obtain by this process such good numerical results as we can get by sub- stituting the numerical values of a 2 , 6 2 , /JL/O? immediately in the equations above. If we do so we get the equations 4-566T2o 1 -2-l7232& 1 + '08093a 3 - 2-14128o 1 -0-995646 1 + '06127a 3 - o 02349^- 030126 1 +12-51451a 3 -6'485086 3 = '^n /2 x 5'00455, o 020420!+ -024066i+ 6-48508a 8 -9'000206 3 =-|n /2 x5-00152. o VIII.] THE PARALLACTIC INEQUALITY. 37 We notice that the first equation is not very different from the second doubled : it is this fact that makes successive approximation a disadvantageous method and renders it ad- visable to include small quantities from the beginning. Eliminate a 3 , b s in succession from the third and fourth equations, thus : Multiply the third equation by 9'00020-r [12-51451 x 9'00020 6'48508x6'48508] = 0*127523, and the fourth by -6'48508---[12'51451x9'00020-6-48508x6-4850S]=-0-09]887, and add ; b s will be eliminated. Again multiply the third equation by '091887 and the fourth by '177317 and add; a 3 will be eliminated. Hence we find 001119^ - -006052^ + a 3 = |n /2 x 1-09776, o - -0014630! - '007034&! + b 3 = f n' z x 1-34669. o Multiply these by '08093 and '05137 respectively and add to the first equation : 5/i' 2 x 2'87937, - 0-00009^ + 0-00049&! - '08093a 3 = ?i' 2 x - 0'08884, o -0-000080!- 0-00036&! +'051376 3 =|w /2 x 0'06918 ; o hence 4-566650! - 2-172196! = ? ri* x 2*85971. o Eliminate c/ 3) b 3 in a similar manner from the second equation ; 2-14128a 1 -0-995466 1 +-06127a 3 --033386 3 =-|n /2 x 0'91416, o -000007a 1 +0-000376 1 --06127o 3 =-^n /2 x 0'06726, o -0'00005a 1 -0'000236 1 + '033386,= - 1 ?i' 2 x -0'04495 ; o 38 LECTURES ON THE LUNAR THEORY. [LECT. VIII. hence 2-14116a 1 -0-995506 1 =-|?i' 2 x 0'93647. o From these equations we find 0, = - f n /2 x 46-4814 =-11392,8, o b, = -lri*x 99-0336 =-'24273,4, o a 3 = |?? /2 x -55042= '00134,9, o b s = |^ /2 x -58209= -00142,7. o LECTUKE IX. THE PARALLACTIC INEQUALITY (continued). LET us now consider the terms in 5tjr which have been left outstanding. Include additional terms Xa 5 cos 5^, \b 5 sin 5^|r in SI, SO, and equate to zero the coefficients of cos 5^, sin 5^ in the differential equations that give SI, SO. We have 25a B + a, - 10 (1 + n') b, - +10(1+ ?i x ) a 5 In these equations substitute Then 5 -f 6n' + ^ ri 2 } a 5 10 (1 + n') b s = -~ lb " lo 40 LECTURES ON THE LUNAR THEORY. [LECT. Eliminate b 5 : - 2n' + \ '") 5 = [(I - 9' - ?f ' + n" + 1 and then 6 5 is given by From these we find = /i /2 x -00595,3 = '00001 4591, o & 5 = JV 2 x -00710,3 = -00001,7410. o These numbers being so small, we see that we may safely ignore, as we have done, their effect in modifying the earlier coefficients. To find the effect of these coefficients upon the Moon's coordinates we must multiply by the factor X= pt~r~jg - "> We shall take in accordance with the results given in Monthly Notices, Vol. 13, p. 177, and Appendix to the Nautical Almanac, 1856, Constant of Moon's Parallax = 3422"'325. Also .we shall take in the first place, the Sun's Mean Parallax to be 8":8, and in the next place 8 // '9, and we will find the corresponding values of the coefficients of the Paral- lactic Inequalities. IX.] THE PARALLACTIC INEQUALITY. We find 8"'8 X = -00250,9 \a, = - -00028,585 41 -00060,903 =-125 // -62 \a s 00000,3385 00000,3580 = Xa 5 = -00000,00366 X& 5 = -00000,00437 = // '7384 0" -00901 8"-9 X = -00253,76 Xax = - -00028,910 X&! = - -00061,596 = -127"-05 Xa 3 = -00000,3423 X& 3 = -00000,3620 0"-7468 Xa 5 = -00000,00370 X6 5 = -00000,00442 0"'00911. These results are very fairly accurate ; but in order to get good values for a x , b lt we were obliged to discuss a t , 6,, a 3 , 6 3 simultaneously. Let us consider the peculiarity of the equa- tions from which this difficulty arose. Following the method of approximation of Lecture VII, if we neglect at first the products of 81, SO, dSl/dt, dS0/dt with the small quantities a 2 , 6 2 , n z , the equations become , Y j- zn -, .3 oi 4- A =0, dtf dt a 3 + F = 0. Now suppose the following is a set of terms that appear in X pi cos (it + 7), in F # sin (i' + y), 82 a; cos (^ + y), 80 &i sin (it + y) ; then as in Lecture VII, we find 2 - + ^ J^EUBR^S^ ^ OF THE **P UNIVERSITY i4UFOR^ 42 LECTURES ON THE LUNAR THEORY. [LECT. IX. Therefore if i differs little from n, the divisor in c^ will be small, and a small error or omision in the numerator of c^ will appear magnified in the values of both ai and &;. In the case of the first term of the Parallactic Inequality, i = n n', -n+Jfi'* = ~2nn' + | w/ *; Zi and if we take _9 /2 _3 /2 Pi 5 " j (i o ^ > o o which differ from the correct values by quantities of the fourth order, then ~ n 3 , 2 5?i 3?i' i 8 n w' and the formulae give _ 3 n' (on - 3n') 4 (n n') (4tt 5ri) ' , 2ti 3 n' 2 Now if we develope these expressions in ascending powers of m, i.e. n'/n, the first terms are a = w & - -i-77 and these are the only terms which the formulae derived from our method of approximation will give correctly. LECTUEE X. THE ANNUAL EQUATION. LET us next take into account the effect of the first power of the eccentricity of the Earth's orbit. We shall find thafc it produces an inequality in the Moon's coordinates, the chief part of which has a period of one year, and is therefore called the Annual Equation. In the formulae of Lecture VII, let the known approximate solutions 1 , # , include the Variation only ; then the equations for the corrections SI, 86 are C S - 2 [(1 + n f ) + 26 2 cos 2^] + 3n /8 (sin 2^+ 6 2 sin 4x/r)S0 = 0, sin 2 + 3n/ * ~ 62 + cos 2 + 6 cos where a 2 , 6 2 , /^/ci 3 are known quantities whose values are given in Lectures IV, V. Refer now to Lecture III, and we find that the terms that are left outstanding when the terms of the Variation are sub- stituted, and the parallactic terms omitted are the following : 44 LECTURES ON THE LUNAR THEORY. [LECT. X = - I n V cos (n't - w') - ^ n V cos (2(0- w'$) - (w'J - w')} 2 4 + | w V cos f 2 ((9 - w'f) + (n't - w')} F= + ^ nV sin {2 ((9 - w'$) - (n't - w')} - | /iV sin {2 (0 - n't) + (n't - w')}. Write a for n' tar' ; then cos }2 (d - n't) - a} = cos (2^ - a) - (26 8 sin 2^) sin (2^ - a) = - 6 2 cos a + cos (2-^r a) -f b 2 cos (4-^r a), sin (2 (0 n't) a) = + 6 2 sin a -I- sin (2-\/r a) + 6 2 sin (4i|r a), cos {2 (0 n't) + a} = 6 2 cos a + cos (2>|r + a) + 6 2 cos (4i|r + a), sin {2 (6 - ft'tf) + a) = 6 2 sin a + sin (2i|r + a) -f 6 2 sin (4\/r + a). Hence q 91 X = - I w /s (1 - 36 2 ) ^ cos a - ^- wV cos (2^ - a) z. 4 3 21 3 '' ' + -j n V cos (2>Jr + a) -^ n' 2 - F = 6n /2 6 2 e' sin a + wV sin (2-f - a) - ?i /2 e' sin (2^/r + a) 21 3 + -j- n' 2 b 2 e' sin (4^ a) - ^i'^e' sin (4-v/r + a). TP T< For our present purpose we shall ignore the small terms in and 4-Jr + a. which are of the sixth order. Assume SI = a 5 e' cos a + a$' cos (2>/r a) -f a 7 e f cos (2ty + a), 80 = b 5 e sin a + 6e' sin (2^ - a) -f 6 7 e sin (2^ + a). Now the terms which arise in the left-hand members of the X.] THE ANNUAL EQUATION. 45 equations owing to terms a p cos pt in SI, and b p sin pt in SO, will be jo% cos pt + 2a 2 pa p [cos(pt 2-^r) cos ( p + 2^)] + ~T a p cos 2^ + ^> a 2 cos ( /> 2-*J a- |_ 2 - 2 (1 + n') A cos pZ 2b 2 pb p [cos (pt - 2-\Jr)+cos(p +2i/r)] + - n'*b p [cos ( pt 2^) cos (pt + 2-\Jr)], and - p*b p sin ^ + 2a 2 pb p [- sin (pZ - 2^/r) + sin (pt + 3 + - n 2 b p [sin (pt 2>Jr) 26 2 sm jci + sm (^9^ + 2 (1 + ri)pa p sin j3^ + 2b. 2 pa p [sin(pt respectively, neglecting the very small quantities in 4-^r. Hence we get the equations following : Equate to zero the coefficients of cos a, sin a : _2rc'(l+O& 5 + 2(2-< -2(2- n') bj) + ri /2 6 6 + 2 (2 + n' - 2 (2 + n') b z b 7 + 1 7/ 2 & 7 = | n' 2 (1 - 36 2 ) - ?? /2 6 5 - 3rt /2 6 2 6 5 + 2(1+ n') ria 5 + 2 (2 - n') a 2 6 6 - 2 (2 - w') 6 2 a 6 - n'*b 6 -2(2 + n' + 2(2 + n' Equate the coefficients of cos (2-^r a), sin (2-*Jr a) : (2 - nja, + 3 3 a 6 - 2 (1 + n') (2 - n') b s + 2n'a0, + 3 ^ 3 . | a 2 a 5 2 (1 + O (2 - n ) a 6 + 2n'aJ> 5 - 2ri 46 LECTURES ON THE LUNAR THEORY. [LECT. Equate the coefficients of cos (2-\Jr + a), sin (2i/r -+ a) : (2 + nja 7 + 3 - 3 a 7 - 2 (1 + n') (2 + ri) 6 7 - 2rc'a 2 a 5 + 3 - ? a 2 a 5 - (2 f w') 2 6 7 - 3>i' 2 &A + 2 (1 + n') (2 +!-*.- !-. In equations of this class, as a general rule we would determine a 5 , b 5 approximately from the first pair, substitute them in the second pair and determine a e , 6 6 approximately, and similarly o^, b 7 from the third pair, and repeat this approxi- mation as often as might be necessary. But if we refer to the second equation, we see that b 5 must be determined by means of a small divisor, and this puts any method of approximation at a disadvantage. In order to obtain readily satisfactory values for the new coefficients, we shall treat the six equations simul- taneously, substituting first the numerical values of the known quantities. We have found a 2 = -00717,95, 6 2 = -01021,20, /*/a 3 = 117150,3. Hence 3-52105a 5 - 0'1747636 5 + -0654050. - '029393& 6 + -067727a 7 - -032695& 7 = I n /2 x 0-969364, 0-l74763a 5 - 0'0067366 5 - '039197a 6 + -0177536 6 + '042499a 7 - -0200756 7 = | n" 2 x - 0*040848, 039009a 5 + '008153& 5 + 7l9766a 6 - 4'148586 6 = n /2 x 3-50, - -00165105- -0086436 5 + 414858a 6 -3-683356 6 = 5 n' 2 x - 3-50, X.] THE ANNUAL EQUATION. 47 036687a :5 - -0114556, + 7-84443a 7 -4'498116 7 = |?/ 2 x-0-50, 001651a 5 + -010965&5 + 4*4981 Ia 7 - 4-3301 26 7 = o' 2 + 0-50. A From the second and third pairs we find 016186a g + '0070846 5 + a 6 = | n'* x 2'94739, fi 018678a 5 + '0103266 5 + b, = | n /2 x 4 26994, A 011026a 5 - -0072036 8 + % = | ?i' 2 x - 0-321398, 011072a 5 - -OlOOlo&g + b 7 = n'* x - 0449338. Eliminate a 6 , 6 6 , a 7 , 6 7 from the first pair. Hence 7i' 2 x Q'909170, 0-l74819cr B - -0065366 5 = | n 2 x 0-003516. Hence a 5 = 1 72 /2 x - 0-70619,8 = - -00692,37, 6 8 = |n' x - 19-4268 = - -19046,3. i Now e' is a -constant found by observation ; taking e' = 3459 // '28, its value in 1850, we get a/ = --0001 1,61, &/ = - -00319,4 = - 658"-9, and further a 6 = -03035,8, a,e' = '00050,9, b 6 = -04396,7, b 6 e'= '00073,73= 152"'09, a 7 = - -00444,7, a/ = - '00007,457, & 7 = - -00623,6, 6 7 e' = - '00010,46 = - 21"-57. LECTURE XI. THE EQUATION OF THE CENTRE AND THE EVECTION. WE have seen that the equations of motion dt 2 dt dt are satisfied very approximately by the values I = log - = a 2 cos 2ijr, a = nt + e -f b 2 sin 2i/r, where ^r nt + e - (n't + e), and a 2 , 6 2 are small quantities depending upon the ratio n'jn, and a is a quantity depending upon % in such a way that while n, e are arbitrary, though subject to the assumption that the ratio n'/n is small. This solution, then, expresses a possible case of motion ; nevertheless it is no more than a particular case because it involves only two arbitrary constants, whereas the complete and general solution must contain four, in order that it may be able to satisfy any given initial conditions, that is, in order that the initial coordinates and their initial velocities may have any given values. LECT. XL] EQUATION OF THE CENTRE AND THE EVECTION. 49 When there is no disturbance the four arbitrary constants are n and e, which denote quantities similar to those expressed by the same symbols above, and the two elliptic elements e and OT, of which e denotes the eccentricity of the orbit and BT the longitude of the apse. We will now shew how to complete the solution by intro- ducing into log (a/r) and 6 additional terms depending on quantities similar to e, or, of which the former is constant and the latter varies slowly and uniformly with t ; and for the sake of simplicity we will suppose at first that e is so small that its square and higher powers may be neglected though it is other- wise arbitrary in magnitude. Let us assume then log - = a 2 cos 2>/r + e cos (nt -cr), 9 = nt + e + 6 2 sin 2*^ + 2e (1 + &) sin (nt - -07), in which the elliptic terms are of the same form as in the undisturbed orbit, and is is supposed to be slowly variable, so that where p is supposed to be a small quantity of the order of the disturbing force. We will now substitute these assumed values in the differ- ential equations. We have -si 2 (n n) a 2 sin 2i|r + (n p) e sin (nt cr), = 4 ( n n') 2 a 2 cos 2^/r -f (n pfe cos (nt -or), dO -J- = n + 2 (n n') 6 2 cos 2^/r + 2 (n p) (1 + 6 ) e cos (nt OT). = - 4 (n - n') b 2 sin 2^/r 2 (n pY (1 + b Q ) e sin (nt - -OT). Hence 4 (n n') 2 a> 2 cos 2-|r -f (n pf e cos (nt w) + 2 (n - n') (n-p) a 2 e [cos (2-^r nt + vr) cos (2i|r + nt &)] A. L. 4 50 LECTURES ON THE LUNAR THEORY. [LECT. - [n* + 4n (n - n') b 2 cos 2i/r + 4Jr - 3 sin 2>Jr [2 (1 + 6 ) e sin (nt - 8r)]| = 0, and - 4 (n - w') 2 6 2 sin 2i|r - 2 (n - _p) 2 (1 + 6 ) e sin (n< - *r) + 4?n (n n') a 2 sin 2-v/r + 2?z (n p)e sin (w r) 4- 4 (w - w x ) (n p) (1 + 6 ) ea >* [sin(2i/r ?i^ + ) 4- sin ( 2i|r + ^ - t + 2 (71 - ri')(7i - p) e6 2 [- sin (2-^r n t + ^)-\- sin (2-^ + nt + 7i /2 g sin 2^ + 3 cos 2^ [2 (1 + 6 ) e sin (w* - r)] = 0. It will of course be found that with the values of a 2 , 6 2 of Lecture IV, the terms independent of e vanish identically. Equating to zero the coefficients of cos (nt TV) in the first equation and sin (nt OT) in the second, we get (n -pf - *n (n -p) (1 + 6,) + '^ = 0, a -2(n- p) 2 (1 + 6 ) + 2n(n-p) = 0. Therefore (n p) (1 + b ) = n, and (n-jo) 2 =4?i 2 -'^ a g = n z <: 7i /2 , approximately, or - = -^m 2 = b 0) approximately. Now terms have been left outstanding with the arguments 2-^ nt + 'sr, 2->/r + nt -ST. These may be removed by assuming log - = Oa cos 2i/r + e cos (nt r) + a 2 i6 cos (2^ wtf + /SJ ") + a 22 e cos (2i|r + nt r), sin 2i/r + 2e (1 + 6 ) sin (nt *r) - nt + r + 6 22 e sin (2r -f n - OT ). XI.] EQUATION OF THE CENTRE AND THE EVECTION. 51 Hence in place of the former equations, we get the following (n - n') (3n - 2' - p) + 0,0. - 2 (w -jp) 2 (l + 6 ) - 2 (w - n') (n - 2ri + p) b t a^ + 2(n- n') (Sn - 2ri - 2 (n - n') (n - 2n' f- 2 (w - w') (3n - 2n' -p)a 9 + n /2 6 22 = 0. Multiply the second by - ? ^-, and add to the first; this will eliminate 1 + 6 . 2 (TO - n') (w - 2w' +p) [> 2 a 21 - + 2(w-TO' 42 52 LECTUEES ON THE LUNAR THEORY. [LECT. Also the equations obtained by equating the coefficients of e cos (2>/r nt + iv) and e sin (2^r nt + -cr) to zero are 2 (n - n') (n -p) + - 2n' + pf + oa - 2n (n - 2w' + p) b zl = 0, 4 (n - n') (n -p) (1 + b ) a 2 - 2 (n - n') (n ->) 6 a - 3^ /2 (1 + 6 ) - (n- - 2w x + jo) 2 6. 21 + 2n (n - 2ri + p) a 2l = 0. 2t? Multiply the second by ^ , ----- , and add to the first ; fl ~~ u ~ this will eliminate 6 2 i> an d gives I - 4 (re - n') (n - p) (1 + 6,) + 4 ^ ^ "^f + ^ (n - p) j 6, Lastly the equations obtained by equating the coefficients of e cos (2-\|r + nt OT) and e sin C2\fr + n -or) to zero are -2(^-^)(^-l>)+|^la 2 -4(^-^)(^- a - 2w (3n - 2n' -p) 6,, = 0, and 6o)os + 2 (w-tt / )(w- - 2w' - p) 0^ = 0. n Multiply the second by ^-7 -- and add to the first ; J 3n-2?i'-p this will eliminate 6 22 , and gives XI.] EQUATION OF THE CENTRE AND THE EVECTION. 53 - 2(n - n')(n -p) + \ - 3n _^ n ,_ p (n - nO(n -p)(l +6,)] a, ?* - 2w' - u 2 - 4n a + a = 0. These six equations are to be solved by successive approxi- mation ; taking the first rough values of p/n and 6 , we find from the last two pairs values for a 2 i, ^21 , a.^, b.^', these are substituted in the first pair and yield more approximate values of p/n and b oy and so on. It will be noticed that this complexity is made necessary by the fact that 02!, & 2 i are found by means of a small divisor LECTURE XII. THE EVECTION AND THE MOTION OF THE APSE. WE proceed to the conversion into numbers of the formulae of Lecture XL Take n n = 1, n = 1-08084,9, n' = "08084,9, ^= 1-17150,3, CL log a 2 = 7-85609, log 6 2 = 8-00911. First Approximation. O-p) 2 -4n 2 + 3^ = 0, cu 4rc 2 4-67293,7, - 3 ^ - 3-51450,9, CL (n-pf= 1-15842,8, n-p= 1-07630,3, p = -00454,6, n-2n' + p= -92369,7, 3n-2n / -jp= 3-07630,3, 1 + 6 - 1 -00422,4 = n/(n - p). LECT. XII.] THE EVECTION AND THE MOTION OF THE APSE. 55 Substitute in the equation for a 21 , 4rc 2 + a 21 + 2 (n - n') (n - p) a, + a, -n')(n-p)^^ +6^ The various terms give 2 (n - n') (n -p) a 2 '01545,45 1^,0, -03784,83. ~ 8 n ^^ ^ ~ W/ ) ^ " P ^ 1 + bo) a2 ~ ' 7264 ' 10 -04415,05 05144,47 ^ /2 (1+6 ) -01969,25 7171/2 " ^ -04608,58 05373,43 85321,6 -4n 2 + 3- } -1-15842,8 a 3 ' 30521,2 a 21 = -17605,6. The equation for 6 21 is - 2 / '\J\-^ ( n ~ P) b * ~ 3 56 LECTURES ON THE LUNAR THEORY. [LECT. l n , a 21 -41201,8 n 2n + p ^-"ol'l^fr -P)(l +&o)o. '03637,95 - -02576,42 6 21 = '39955,3 The equation for o^ is Here - 2 (n - w') (w - p) a 2 - '01 545,45 |^a 2 -03784,83 6o)a2 -* 02181 ' 13 - 4 (n - w') (w - p) (1 + 6 ) 6 2 - -04415,05 - 4 ^ (n - n') (n -p)b, - '01544,69 3n 2n p -3w /2 (l-l-6 ) --01969,25 - 6 =- -^- f (1 + 6 ) - '01383,78 - -09254,52 9-46363,4 - 1-15842,8 8-30520,6 = 01114,30. XII.] THE EVECTION AND THE MOTION OF THE APSE. 57 And 0783 ' 011 + 6o) " 2 '00327,988 - 00232 ' 283 M ' 0208 ' 086 = -01551,37 Second Approximation. The complete equation for ^> is 2 (n - w 7 ) (w - 2n' ^ /2 [^21 + 6 J + 2 (w - w') (3n - 2w' - 4 (n w') (3w - 2r&' p) [62^22 - 1-15842,8 00080,8 - -00520,1 58 LECTURES ON THE LUNAR THEORY. [LECT. -00708,4 -00406,9 & 2 & 22 ] - '00048,2 (?? - n') (n - 2n' + p] [6 2 a 21 - a 2 6 21 ] - '00397,5 - 4 -- (n - w')(3w - 2' -p) [6^ - a 2 6 22 ] - '00003,0 3-[6 21 -6J -00756,1 (n-pf= 1-14859,4 n-p = 1-07172,5 p= -00912,4 p : n = -00844,2. Apply these numbers in the equation for 1 + 6 : i - b J 4(w-|, 1-00851,33 - -00017,51 00086,55 (3n - 2' - p) [6,^ - Os&J -00000,64 "- -00163,92 1+6 = 1-00757,1 Continuing the approximation for 21 , 6 a , a^, ^ the various terms found are the following : XII.] THE EVECTION AND THE MOTION OF THE APSE. 59 01538,88 03784,83 - -07221,53 - -04410,93 05097,33 01975,81 04601,14 divisor 86169,5 - 1-15842,8 -01538,88 divisor 03784,83 9-43549,4 -02182,34 -1-15842,8 - '29673,3 - -04410,93 8-27706,6 Oa = -18082,0 42108,0 03598,79 -02540,22 - -02292,94 -01540,41 - -01975,82 - -01390,46 a.x = -01118,03 05365,53 - -09254,01 00786,805 00328,659 00231,985 00209,403 40873,6 6 ffl = -01 556,85 Third Approximation. We find 1-15842,8 - -00081,40 00535,02 - -00723,54 - -00416,02 00048,35 00407,40 00002,97 - -00777,57 (n-p)*= 1-14838,0 n-p = 1-07162,5 p= -00922,4 p : n = '00853,5 01538,73 divisor 03784,83 -86188,04 07220,54 - 1-15842,8 04410,81 - -296548 05096,30 01975,95 04600,96 05365,42 a 2l = '18092,9. 1-00860,75 - '00017,57 00087,94 00000,64 - -00167,84 1-00763,9 -01538,73 divisor 03784,83 9-43488,0 -02182,35 -1-15842,8 -04410,81 8-27645,2 -01540,32 -01975,95 -01390,60 -09253,93 a SB ='01118,10. 42128,90 03597,92 - -02539,43 - -02292,60 = '40894,8 00786,881 00328,671 00231,978 00209,430 = -01556,96 60 LECTURES ON THE LUNAR THEORY. [LECT. XII. Fourth Approximation. 1-15842,8 1-00862,79 - -00081,41 - -00017,57 00534,22 n-p= 1-07160,3 '00088,04 - -00727,13 p= -00924,6 -00000,64 - -00416,24 p:n= '00855,4 -^0167^94 '00048,35 i + 6 = 1-06766,0" 00407,66 00002,97 - -00778,05 (n-p?= 114833,2 The values already found for the remaining quantities are sufficiently exact. These numbers give, taking after Hansen, e(l + 6 ) = '05491 e = -05449 2 (1 + 6 ) e = -10982,0 = 22651 // '9 -00986,03 -02228,44 = 4596"'6 a 22 e = -00060,93 b^e =-00084,85= 175"-1, and taking the Moon's mean annual motion 17325593", the annual motion of the apse is 148202 // = 4110'2 // . LECTURE XIII. THE MOTION OF THE APSE, AND THE CHANGE OF THE ECCENTRICITY. WE have seen that when the eccentricity of the Moon's orbit is not considered we may write r a H = no? [1 + As cos 2(0- 0')], 3 2-m 1 ,3m 2 where a 2 = ^ m . :, . .. ., ; ft 2 = -r- ., . 2 1 -m , 11 . 4 1 m 3 - 8m + -g- m 2 Let us introduce the two new arbitraries e, TV by writing H = hna? [1 + h 2 cos (2 - 2m) 0], - = ~- [1 + a 2 cos (2 - 2m) + e cos (0 - )], where /i is a third arbitrary, which may be chosen to suit our convenience ; it must be unity when e = 0. Then dH dH dt 3 [sin (2 - 2m) + 4me cos (2 - 2m) sin (0 - tar)] - 1 m 2 ^ - 1 m 2 n 2 f^ [1 - (4oa + Aa) cos (2 - 2m) (9 - 4e cos (0 - CT)] x [sin (2 - 2m) + 4me cos (2 - 2m) sin (0 - r)], 62 LECTURES ON THE LUNAR THEORY. [LECT. and also 7 TT 77 dO = d6 n0? t 1 + ^ 2 C S ^ 2 ~ 2m ^ ~ na2h ( 2 - 2m ) ^ sin(2 - 2m) (9. Now we may put h 1 + 77, where ?; vanishes with e. Neglecting powers of e above the first drj dh ^ = ^ = 3m 2 (1 -f m) e sin (1 - 2m(9 + -or) du du + 3m 2 (l-m)esm (3 - 2m0 - r) - 9m 2 77 sin (2 - 2m) ^. Neglect at first the last term : Substitute this in the last term, and we get 27 2 + 4m -4m 2 ( Now consider the other equation H 2 LL 1 m2wV [ cos ( 2 - ' - 2me cos (1 - 2m0 + w) + 2me cos (3 - 2m0 - )], Differentiate the assumed expression for -, and let ^ be chosen so that the first differential coefficient shall have the same form as if h, e, vr were constant. Thus 1 dr I where XIII.] THE MOTION OF THE APSE, ETC. 63 or cos (d tff) + e , sin(0 - w) = 6m 2 (I+m)e sin (1 - 2m0 + + 6m 2 (l-m)e sin (3 - 2m0 - ) and = - (2 - 2m) a 2 sin (2 - 2m) 4- - e sin (0 - 77^7 = -j- (2 - 2m) a 2 sin (2 - 2m) + ~ [1 + A 2 cos (2 - 2m) 0] e sin (0 - w), nadh , . d6 Multiply by r*/n?a 3 ; then since r 2 -5- = H = /ma 2 [1 + / 2 cos (2 ttc we have ^.2 x72,v. __ i = (2 - 2m) 2 a 2 cos (2 - 2m) / Cv Cviu - (2 - 2m) A 2 sin (2 - 2??i) e sin (0 - r) + e cos (0 - tsr) + 2A 2 cos (2 - 2m) e cos (0 - (2 - 2m) a 2 sin (2 - 2m) 64 LECTURES ON THE LUNAR THEORY. [LECT. and this is equal to 7T2 .. 1 3 O ~3 T + ^m 2 -, 4- ^m 2 -, [cos (2 - 2m) (9 n 2 a?r rc 2 a 3 2 a 3 2 a 3 L v - 2me cos (1 - 2m0 + -BT) + 2me cos (3 - 2ra0 - r)] -Tl + \ m ^ C 1 ~ 3a cos ( 2 " 2m )^ ~ 3e cos (^ ~ OT )] 1 m 2 A 6 |cos (2 - 2m) + 6a 2 e cos (6 - r) ) e cos(l-2m^ + w)- (| - 2ml e cos(3-2m^- r) 1 - 2m The terms in these two expressions which are independent of give no new information ; equating the others : - (2 - 2m) htf sin (2 - 2m) sin (0 - ) (2 - 2m) a 2 sin (2 - 2m) <9 o 3 r = 3m 2 ?; - -= m*e cos (6 - r) + ^ m 2 6a 2 e cos (0 - r) -] e cos ("l^2m0 + w) - (^ - 2m^ e cos (3^2w0-r) - + 2m- e cos (l + 9m 2 7? cos (2 - 2m) 0. Keducing this expression (q gQ \ ^ m 2 + 3m 3 + g- m 4 J e cos (1 - 2m0 + r) s m 2 - 3m 3 + > 4 e cos (3 - 2m0 - r) OF THE UNIVERSITY OF XIII.] THE MOTION OF THE APSE, ETC? " 65 and from before ,-A cos (6 -ay) + e -T5 sm (0 -BT) = I8m 4 e sin (0 OT) du CLu + (6m 2 + 6m 3 ) e sin (f- + (6m 2 - 6m 3 ) e sin (3 - 2m.0 -F r). Hence, de ! 3 - - m m 3 m2+ 3 m3+ 15 We notice that among these terms, one is of long period, approximately semiannual, and will become of greater relative importance than the others on integration. To effect this integration, assume n = tsr-f a sin 2 (0 - w)+ ft sin (2 -2m) 6 + 7 sin {(4 -2m)0-2'&}, so that the mean motion of II is the same as that of &, and substitute in the equation. A. L. 5 66 LECTURES ON THE LUNAE THEORY. [LECT. Then x cos 2 (0 - -cr) [q on g | m 2 - 6m 3 + ^- m 4 f (2 - 2m) - 6m 2 a - ^ m 2 ? x cos (2 - 2m) f ^ m2 + 1 m 3 + 6| m 4 _| m2 J cos ( 2m(9 - 2H) -^ m 2 +^m 3 + jg m 4 + (4-2?tt)7-^m 2 a -^w 2 7 x cos (4- so that if we take 219 a = _ - m + 8 32 9 2 15 ,_99 4 3 3 51 we have j TT o Q O Q (~ 1 P\ Q 4< \ ~1 ^ = ^ m -2 + ^ m 4 + M^ m 2 + | m 3 + ^ m 4 cos ( 2m (9-2n). If we write mB II = i/r, this becomes -^ = a b cos 2i|r, where 3 309 ,15 9 45 a = m - - m 2 - -^- m 4 , 6 = -7- ?^ 2 + ra + -g- m 4 , and the solution is a + 6 tan- 1 A -, tan = 0\/a 2 - 6 2 + constant. . ' XIII.] THE MOTION OF THE APSE, ETC. 67 Hence if we denote by -~ the mean rate of change of cr, we have dS 3 225 4071 ^ (vyi _ ntYi" - - nm^ - - 100* 4 W 32 rt 128 m ' c^ _ 3 225 4071 dB ~ 4 h ~32 * h 128" We observe that a -f 6 and a 6 are the rates of separation of the Sun from the apse when the Sun and the apse are at quadratures and syzygies with one another, respectively, that is if we take II for the longitude of the apse, or, what is the same thing, if we ignore small terms of short period. Hence the mean rate of separation of the Sun from the apse is a mean proportional between its rates when at quadratures and syzygies respectively with the apse*. [* This is the analogue for the case of the apse of Machin and Pemberton's theorem on the motion of the node, inserted in the third edition of the Principia as a scholium to prop, xxxiu., lib. in. See some notes by Adams in Brewster's Life of Newton, Appendix xxx.] 52 LECTURE XIV. THE LATITUDE AND THE MOTION OF THE NODE. LET us first treat this problem on the supposition that the latitude is so small that its square may be neglected. The equation of motion, taken from Lecture II, may be written d*z z mfr /- E - M r where z = r sin (latitude) and the cube of s is omitted; or neglecting the parallactic terms ^__J/f + ^l dV~ [r* r' 3 J' The value of /i,/r 3 may be considered known by the operations which have determined the motion in an orbit coinciding with the ecliptic ; that is to say, = ~ [l + 1 2 2 + 3a 2 cos 2^ + (J a 2 2 + 3a 4 ) cos where a has the definition of Lectures IV, V ; or numerically, taking n n'= 1, ^ = 117150,3 + -02523,0 cos 2t + '00025,15 cos 4>t. And ^ = rc /2 = '00653,6. Hence Jv ~ 2 = - z [1-17803,9 4- -02523,0 cos 2t + '00025,15 cos 4ft]. LECT. XIV.] THE LATITUDE AND THE MOTION OF THE NODE. 69 Let us now consider the equation where P = q + 2g x cos 2t + 2g 2 cos 4ft, in which q lt g 2 are supposed small. Suppose a term in z to be c cos (to + /3) ; when this is substituted in Pz there will arise terms c cos (k - 2t 4 ) c cos (& + 2 + ft) c cos (A; - 4t + @) c cos (k + Let us therefore assume z = c [cos (to + ) + Ci cos (A; + 2t + /9) + c 2 cos (k + 4 + ft) + c_! cos (A; - 2t + yS) + c_ 2 cos (k - c is arbitrary ; we have to determine k, c 1? c_i, &c. Substitute and equate coefficients: =0 =0 =0 ... =0 If ^D ^2v are neglected, we have simply - & 2 + q = ; this is a first approximation to the value of k. Taking q 1 into account and neglecting q 2 70 LECTURES ON THE LUNAR THEORY. [LECT. In the actual case considered we notice that g does not differ widely from unity. Hence k is nearly equal to unity also, and the denominator in c_i is small, and makes c_j much more important than c l . If we substitute these values in the third equation above, we have whence (P - g ) 3 - 8 (& 2 - qrf - {16 (g - 1) + 2^} (k* - q) - Sqf = 0, which may be put under the form (k* - g ) 2 + 2 (q - 1) (k* - q.) = - qf + i q? (k* - g ) + | (A; 2 - g ) 3 , whence (f - 1) 3 = (?<, - 1) 2 - g, 1 - 1 j- 3 cos &) x (sin 6 cos & cos 6 sin & cos i) MG in the plane of the orbit, , perpendicular to TJ- 3 cos ft) x sin 0' sin i the plane of the orbit. XV.] MOTION IN AN ORBIT OF ANY INCLINATION. 75 Now cos a) = cos cos & -f sin sin & cos i = cos (6 - 0') cos 2 \ + cos ((9 + 0') sin 2 ^ , L 2t sin # cos #' - cos sin 0' cos i = sin (0 - 0') cos 2 ^ + sin (6 + (90 sin 2 \ . - _ Hence we have the following expressions for the three forces : x 2 cos 2 sin2 + ^ + cos 2 ^ " \ 7? sin * - sin (0 - 20') cos 2 - + sin cos i + sin (0 + 20') sin 2 ^ Now we have seen dNZrsinO _^ __ dt" H ' ~dt = ~ Hsini' also, the rate of advance of the node along the orbit is Zr sin H tan i ' Thus the equations of motion become r 2 ~ dt together with _ dt LECTURE XVI. MOTION IN AN ORBIT OF ANY INCLINATION (continued}. To satisfy the equation at the end of Lecture XV, assume neglecting the square of the disturbing force and the eccen- tricity; thus in the small terms we write d0 dO' r = a, Tt =n, r=a, ^--n. Hence d?r ~d^ = n ' a ^ 2 " 2m)2 Al C S 2 ( 6 " ^ + 4 ^ 2 cos 26> + 4m 2 A cos 20' + (2 + 2m)' 2 ^ 4 cos 2 (0 + 0')] ; substitute in the equation therefore 4- A 1 cos 2 (6> - 6') + ^ 2 cos 2(9 + 4, cos 2<9 7 + At cos 2 (0 + 0')] 4 ["{1 + cos 2 - ^ cos 4 - - w 2 a 4 [(2 - 2m) 2 A cos 2 ((9 - 6> 7 ) + 4>A 2 cos 2(9 8 cos 2(9' + (2 + 2w) a ^ 4 cos 2 (0 + 0')]. LECT. XVI.] MOTION IN AN ORBIT OF ANY INCLINATION. 77 Again, we have the equation =- dt which may be written H ~ = - 1 n'mW [~sin 2 (6 - 0') cos 4 1 + sin 26 . 2 sin 2 1 cos 2 1 + sin 2 (0 + 0') sin 4 ^ 1 . Substitute for pa its approximate value ttV in the small terms ; and we find from these two equations - (1 - m) A l + 4 (1 - m) 3 A l + = m? (1 - m) cos 4 1 = - - m 2 cos 4 1 , - J. 2 + 4^1 2 + ^ m 2 . 2 cos 2 sin 2 ^ _ & A 3 i' i = TC m 2 . 2 cos 2 - sin 2 ^ , 1 22 3 i i - - mJ.3 -H 4tm*A s + Jr m 3 . 2 cos 2 ^ sin 2 x = 0, Therefore _3 2 4 i_ 2-m A z =- 2 m 2 cos 2 ^ sin 2 -, ^ 2 2 3 . .i 2 + m 2'" " 2(l+7n)(l+2m)(3 + 2m)' and H* = nW 1 + m 2 - |m 2 ("cos 4 | + sin 4 |) cos 2 ^ - & + 3m COs2 Sin2 COS 78 LECTURES ON THE LUNAR THEORY. [LECT. where a is defined by If we preferred to define a so that the constant term in H* were equal to n*a 4 , we should have w 2 a 4 = fjua -f m?n?a 4 = m 2 ?i 2 a 4 (cos 4 = + sin 4 =r] , 2 \ 2 2J F 1 i i~\ or fjL = n 2 a 3 1 + <> m 2 - 3m 2 sin 2 ^ cos 2 . Let us next find the latitude and the motion of the node. Suppose that i = i + At, in which Ai, AJV are small, i is a constant, and N varies slowly in proportion to the time, so that we may assume = N-L sin 2 (0 0') + N 2 sin 20 + N a sin 20' -f- J\ r 4 sin 2 (0 4- 00, Ai = /a cos 2(0- 0') + / 2 cos 20 + /, cos 20 X + 7 4 cos 2 (0 + X ). Then remembering that rf0' = _dN an expression that must be used in the terms of chief import- ance, we have =_2(l-m)n/ 1 sin 2 (0 - 00 - 2w/ a sin 20 at _ 2 m - nI 3 sin 20' - 2 (1 + m) n/ 4 sin 2 (0 + 7 ), 71 + 2 fm - - ^ w^ 3 cos 20'+ 2(1 + m)nN 4 cos 2(0+0') \ n at J s di . nnf + ~T- T. sm 20 7 , c?i cfa in which the last term will be found to be required to get the constant q correctly to the order m 3 . XVI.] MOTION IN AN ORBIT OF ANY INCLINATION. 79 These must be equated to Zr cos _ Zr sin ~1T~ 5 TT^inT respectively. Hence q 2mN 3 2 mI 3 ~ = -r m 2 cos i ; O/l T* therefore as a first approximation g g = ^r m 2 cos i ; hence io - 3 3 2/2 = - m 2 sin i cos i, / 2 = ~ m 2 sin i cos i, 9 I _i_ " 2 ' \ T " S * * T " 3 . ~ A T 3 2 .. r_3 m sin t T^ sin t, 2^ TT " ~ > 1 4- T ?ft cos i 4 o/i 3 ...'..* r 3 m 2 . . . ,i 2 (1 + m) / 4 = - m 2 snu sm 2 ^, / 4 = - smi sm 2 and 3 m 2 2 N a and we get the second approximation to q, 39 9 q = '2 m 2 cos i ^ m 2 cos 2 i + -^ m* sin 2 i. 80 LECTURES ON THE LUNAR THEORY. [LECT. XVI. It will be observed that / 3 , ^ are of lower order than the other coefficients, so that in order to obtain them correctly to the same order as the others we were obliged to retain small terms in -=- arising from the variability of N. If we take the variable plane defined by the longitude of the node N and the inclination i as the plane to which the position of the Moon is referred, we have the latitude of the Moon above this plane = Ai sin AxV sin i cos 3 ....... r 1 m sin i cos 2 = 8'" 2,3 ."1-m 1 1 + j m cos i sin (6 -20') 5 m 2 sin i cos i sin 6 o sn LECTURE XVII. ON HILLS METHOD OF TREATING THE LUNAR THEORY. LET us suppose the Moon to move in the plane of the ecliptic, and let us refer its motion to rectangular axes in rotation, the rotation being such that the axis of x passes always through the mean position of the Sun; that is, the axes rotate with angular velocity ri, and if we suppose the Sun describes a circular orbit about the origin, its coordinates are x' = a, y' = 0. Let x, y be the coordinates of the Moon. Then the disturbing forces of the Sun upon the Moon relative to the Earth are ra' x a' m' m y ~7~~P~~~^' ~7*~P parallel to the axes of x and y respectively, where p> = (x-aj + y\ and the forces of the Earth on the Moon relative to the Earth are where r' 2 = x* + y\ Now these forces may be written dx' dy' where f> LL m' mx r p a A. L. 82 LECTURES ON THE LUNAR THEORY. [LECT. Hence IL m! f 1 A m' / 3 \ We have tacitly assumed the origin to be at the centre of the Earth ; if we prefer to place it at the centre of gravity of the Earth and Moon, the necessary change is effected by multiplying the last terms, which correspond to the Parallactic Inequalities, Equating these forces to the accelerations of the Moon parallel to the coordinate axes, we have the equations of motion in the form d?x _ ,dy ,, dl 2ri -f - ri*x = -=- dt 2 dt dx ' d*y , ,dx , 9 dl -J; + Zn- sr n>y = Ty , or, as they may be written, d?x _ , dy _ dR dt 2 dt dx ' , -/_ dt 2 "* dt ~ dy ' where R = H + ^ ri' 2 ( 2 + y 2 ) y 2 a V 2 ^ Now suppose we have found values of as and y which satisfy this pair of equations and which involve two arbitrary constants. This may be accomplished by taking assumed developments x = 2&i cos i (t + 7), y = S6f sin i (t + 7), substituting in the equations, and equating coefficients of the various terms. The solution found will include the Variation and the Parallactic Inequalities. Let it be required to amend this solution by the introduction of the remaining two arbitrary constants that are required for a complete solution. xvii.] ON HILL'S METHOD OF TREATING THE LUNAR THEORY. 83 Let the additional terms that we seek be Sx, Sy, which we shall suppose so small that their squares and products may be neglected, let us consider first the terms which are multiplied by the first power of one of the new arbitraries, the original particular solution corresponding to the case in which this arbitrary is zero. Then &x, By are determined by the equations d 2 R d*R _ d 2 R dt- ' dt dxdy" ^ df " where X, Y are supposed known functions of x, y or of t, and have been added here to include disturbing causes not allowed for in the above form of R. Multiply the original equations by -j- , - and add : d?x dx d*y dy _ dR dx dR dy WdiWdi = d^~di + dy^i = dR since a?, y are the only functions of t that R involves ; whence \dt) \dt) ~ ' where C is an arbitrary constant ; this is the integral known as Jacobi's Integral. Let us write then we have F 2 = 2R + G; and from the original equations themselves dV dR dR , = , cos (f> + -j- sin ; dx dy 84 LECTURES ON THE LUNAR THEORY. [LECT. XVII. and dy . * + *. dR . dR And from these, differentiating and substituting for ~ at we get d?V d<> d<> A , d'B dV LECTURE XVIII. ON HILL'S METHOD OF TREATING THE LUNAR THEORY (continued). THE equations for &, 8y are ~da?* X + dxJy 8 ' J + X> the equations for x, y are , j + ztt TT = j j- &x + -j- 9 ov + F; eft _9 >dy _dR ~~~ *""' Multiply the former pair by -7- , -^ respectively, and the latter pair by -, -, and add all together; we get dy d 2 Sy dx d*R dy\ , ( d*R dx d 2 R dy di + dxdy dt) d> f (dtsdy Tt + ~df~dt dRdSx dRdBy y dx dy h dx dt + dy dt ' * dt- dt* d*R dx d*R dy ^ d dR dx* dt dxdy dt ~ dt dx' d*R_ dx d?R dy _ d dR dxdy dt df dt ~ dt cty ' 86 LECTURES ON THE LUNAR THEORY. [LECT. Then our equation may be integrated dx dSx dy dSy _ dR ~ dR ~ dt~dt + dt~dt~~dx t f d y r so that T is a known function of t, which involves an arbitrary constant. Now let us assume &E = v cos w sin , Sy = v sin (fr + w cos <. , .. T ,. dx dy dSx dSy Substitute above for - t ^ t t J. ; we find , -T. - w . = - cos + rf^ . ,\ -T- sin (/> 0?V / ( dR . dR ,\ rr j- sin (f> + r cos 6 } w + T. \ dx d ^ ) r dy dR dR . dV But -- cos + -=- cos 6 = r ( -3r + rfa? ^ \dt Therefore F *- -. + y- = I TT (^ + ^'j w ^ + I whence An arbitrary constant is included on the right. This equation shews that when w is known, v can be found; it remains to determine w. Now by actual differentiation dv dd> dv ____ _ .xviii.] ON HILL'S METHOD OF TREATING THE LUNAR THEORY. 87 Also multiplying the differential equations for Sx, By by sin (j>, cos + F cos <. Substitute and we find C S . - . -r- sm 2 2^ j-sm d> cos c?^ 2 ^rfy X sin + F cos . Now we have seen dto v c?F /^ cos cf) + r : cos 2 d) L u,*, ^c?y 2/ Xsin + Fcos<^>. 88 LECTURES ON THE LUNAR THEORY. [LECT. XVIII. But by the equations proved at the end of Lecture XVII, the terms in v cancel one another, and we are left with the equation for w : y - - -T-; dt/ dt dx* d 2 R . d*R .1 Z -= T- sin -- 5 cos 2