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BOUGHT WITH THE INCOME 
 FROM THE 
 
 SAGE ENDOWMENT FUND 
 
 THE GIFT OF • 
 
 1891 
 
 A.HHfc.i.a 9.Ult..^^ 
 
AN ELEMENTARY 
 TREATISE ON THE LUNAR THEORY. 
 
CAMBRIDGE : 
 PRINTED BT W, METCAUE AND SON, TRISITT STREET, 
 
AN 
 
 ELEMENTARY TREATISE 
 
 ON THE 
 
 LUNAB THEORY, 
 
 A BRIEF SKETCH OF THE HISTORY OP THE PROBLEM 
 BEFORE NEWTON. 
 
 BT 
 
 HUGH GODFRAY, M.A. 
 
 OF ST. JOHN'S COLLEGE, CAMBRIDGE ; 
 LATE MATHEMATICAL LECTURER AT PEMBROKE COLLEGE. 
 
 FOURTH EDITION. 
 
 gonku nnb pfto fork: 
 
 MACMILLAN AND CO. 
 
 1885. 
 
PKEFACE TO THE FIRST EDITION. 
 
 Op all the celestial bodies whose motions have formed 
 the subject of the investigations of astronomers, the Moon 
 has always been regarded as that which presents the 
 g;reatest difficulties, on account of the number of inequali- 
 ties to which it is subject; but the frequent and important 
 applications of the results render the Lunar Problem one 
 of the highest interest, and we find that it has occupied 
 the attention of the most celebrated astronomers from the 
 earliest times. 
 
 Newton's discovery of Universal Gravitation, suggested, 
 it is supposed, by a rough consideration of the motions of 
 the Moon, led him naturally to examine its application to 
 a more severe explanation of her disturbances; and his 
 Eleventh Section is the first attempt at a theoretical 
 investigation of the Lunar inequalities. The results he 
 obtained were found to agree very nearly with those 
 determined by observation, and aflforded a remarkable 
 confirmation of the truth of his great principle; but 
 the geometrical methods which he had adopted seem 
 inadequate to so complicated a theory, and recourse has 
 been had to analysis for a complete determination of the 
 disturbances, and for a knowledge of the true orbit. 
 
VI PKEFAOE. 
 
 Tlie following pages will, it is hoped, form a proper 
 introductiou to more recondite works on the subject: the 
 difficulties which a person entering upon this study is most 
 likely to stumble at have been dwelt upon at considerable 
 length, and though different methods of investigation have 
 been employed by different astronomers, the difficulties met 
 with are nearly the same, and the principle of successive 
 approximation is common to all. In the present work, 
 the approximation is carried to the second order of small 
 quantities, and this, though far from giving accurate values, 
 is amply sufficient for the elucidation of the method. 
 
 The differences in the analytical solutions arise from 
 the various ways in which the position of the moon may 
 be indicated by altering the system of coordinates to which 
 it is referred, or again, in the same system, by choosing 
 different quantities for independent variables. 
 
 D'Alembert and Clairaut chose for coordinates the pro- 
 jection of the radius vector on the plane of the ecliptic and 
 the longitude of this projection. To form the differential 
 equations, the true longitude was taken for independent 
 variable. 
 
 To determine the latitude, they, by analogy to Newton's 
 method, employed the differential variations of the motion 
 of the node and of the inclination of the orbit. 
 
 Laplace, Damoiseau, Plana, and also Herschel and Airy, 
 in their more elementary works, have found it more con- 
 venient to express the variations of the latitude directly, 
 by an equation of the same form as that of the radius 
 vector. 
 
 Lubbock and Pontecoulant, taking the same coordinates 
 of the Moon's position, make the time the independent 
 
PREFACE. VII 
 
 variable ; and when it is desired to carry the approximation 
 to a high order, this method offers the advantage of not 
 requiring any reversion of series. 
 
 Poisson proposed the method used in the planetary 
 theory, that is, to determine the variation in the elements 
 of the Moon's orbit, and thence to conclude the corre- 
 sponding variations of the radius vector, the longitude, 
 and the latitude. 
 
 The selection of the method followed in the present work, 
 which is the same as that of Airy, Herschel, &c., was made 
 on account of its simplicity ; moreover, it is the method 
 which has obtained sanction In this University, and it is 
 hoped that it may prove of service to the student in his 
 reading for the examination for Honours. In furtherance 
 of this object, one of the chapters (the sixth) contains the 
 physical Intrepretation of the various important terms in 
 the radius vector, latitude, and longitude.* 
 
 The seventh chapter, or appendix, contains some of the 
 most interesting results In the terms of the higher orders, 
 among which will be found the values of c and g com- 
 pletely obtained to the third order. 
 
 The last chapter is a brief historical sketch of the Lunar 
 Problem before Newton, containing an account of the dis- 
 coveries of the several inequalities and of the methods by 
 which they were represented, those only being mentioned 
 which, as the theory has since verified, were real onward 
 steps. The perusal of this chapter will shew to what extent 
 we are indebted to our great philosopher ; at the same 
 time we cannot fail being impressed with reverence for the 
 
 * See the Report of the "Board of Mathematical Studies" for ISoO. 
 
Vlll PREFACE. 
 
 genius and preservance of the men who preceded him, 
 and whose elaborate and multiplied hypotheses were in 
 some measure necessary to the discovery of his simple 
 and single law. 
 
 I take this opportunity of acknowledging my obligations 
 to several friends, whose valuable suggestions have added 
 to the utility of the work. 
 
 HUGH GODFRAT. 
 
 St. John's College, Cambridge, 
 April, 1853. 
 
 PREFACE TO THE THIRD EDITION. 
 
 In the present edition several important alterations have 
 been made, many of the parts have been entirely re-written, 
 and a general re-arrangement of the whole subject has 
 taken place. In Chap. iv. Sect. 3, the necessity for the 
 computation of the elements in the order s, u, 6 is shewn. 
 In the Quarterly Journal, Mr. Walton pointed out the 
 logical necessity for having the value of u completely correct 
 to the second order before proceeding to find 0, although 
 the result is the same as when the preparation for both 
 values is carried on simultaneously. I have here proved 
 that there is the same necessity for making the value of s 
 precede that of u, although, again, the change has no effect 
 on the result. 
 
 The term of the third order " Parallactic Inequality " will 
 be found completely investigated in the appendix ; and at 
 the end of the volume a selection of questions from University 
 and College Examination Papers has been introduced. 
 
 August, 1871. 
 
CONTENTS. 
 
 CHAPTER I. 
 
 INTRODUCTION. 
 
 Arts. Pagt! 
 
 1 Newton's Law of Universal Attraction . , .1 
 
 2—5 Principle of Superposition of Small Disturbances . . 4 
 
 6-8 Attractions of Spherical Bodies . . . .6 
 
 CHAPTER n. 
 MOTION KBLATIVE TO THE EARTH. 
 
 9-13 Problem of Two Bodies .... 3 
 
 14-18 Problem of Three Bodies . . . . .13 
 
 15 Path of Centre of Gravity of Earth and Moon, Ecliptic . 14 
 
 17 Errors in the Sun's assumed place , . . ,16 
 
 CHAPTER III. 
 
 DIFFERENTIAL EQUATIONS. 
 
 19, 20 Formation of the DifEerential Equations . . 19 
 
 21, 22 Orders of the Small Quantities introduced . > .21 
 
 23, 24 Values of the Forces P, T, S . . . 23 
 
 25 Order of the Sun's Disturbing Force . . . .26 
 
 CHAPTER IV. 
 
 INTEGRATION OF THE DIFFERENTIAL EQUATIONS, 
 
 2S General Process described .... 2" 
 
 27-30 Terms of a higher order which must be retained . . -8 
 
 32-36 Solution to the first order .... •'- 
 
 37—42 Introduction of c and p . ■ . ;U 
 
X CONTENTS. 
 
 Arts. Page 
 
 43, 44 Connexion between 6" and 6 . . . . 39 
 
 45—49 Solution to the second order — Preliminary remarks . • 40 
 
 50, 51 To compute s to the second order .... 44 
 
 52 To compute u to the second order . . . .46 
 
 53, 54 To compute t to the second order ... 48 
 
 65—57 Coordinates expressed in term of the time . . .52 
 
 58 Parallax of the Moon ..... 54 
 
 59-61 Indication of method for higher approximations . . i^- 
 
 CHAPTER V. 
 
 NUMERICAL VALUES OF THE COEPPICIENTS. 
 
 63 First Method. Theoretical, Values ... 57 
 
 64 Second Method. Values deduced by the Solution of a large number 
 
 of simultaneous Equations . . . . ii. 
 
 €5-69 Third Method. Independent determination of each Coefficient 58 
 
 CHAPTER VI. 
 
 PHYSICAL INTERPRETATION. 
 
 70-83 Discussion of the terms in the Moon's Longitude . . S3 
 
 84—89 Discussion of the terms in the Moon's Latitude . . 74 
 
 90-95 Discussion of the terms in the Moon's Radius Vector . 78 
 
 96 Periodic time of the Moon . . .81 
 
 CHAPTER VII. 
 APPENDIX. 
 
 98 The Moon is retained in her orbit by gravity . 84 
 
 99 The Moon's orbit is everywhere concave to the Sun . . 85 
 100, 101 Effects of Central and Tangential Forces separately considered . 87 
 102-104 The values of g and c to the third order ... 88 
 105-107 Parallaotic Inequality . . . . .91 
 108, 109 Secular Acceleration ..... 95 
 110-111 Inequalities depending on the figure of the Earth . . 96 
 
 112 Perturbations due to Venus .... 97 
 
 113 Motion of the Ecliptic . . .98 
 
CONTENTS. XI 
 
 CHAPTER Vni. 
 
 HISTOKT OF THE LUNAR PROBLEM BEFORE NEWTOK. 
 
 Arts. Page 
 
 114-117 Deacription of the Exoentric and the Epicycle . . 102 
 
 118 Hlpparchiu's mode of representing the Motion of the Apse . 105 
 
 119 Substitution of the Elliptic for the Circular Orbit . . 106 
 
 120 Ptolemy's discovery of the Erection . . . 107 
 121, 122 His manner of representing it . . . . 108 
 123—125 Copemicus's Hypothesis for the same purpose . . 110 
 
 126 Souliaud, D'Arzachel, Horrocks consider it in a different manner 112 
 
 127, 128 Tycho Brahe'a discovery and representation of the Variation . 114 
 
 129 Tycho Brahe's discovery and representation of the Annual Equation 115 
 
 130 Tycho's Table for the Reduction . . . .116 
 
 131 Inclination of the Koon's orbit and motion of the Node calculated 
 
 byHlpparchus ..... 117 
 
 132 Tycho BrahS's discovery of the change of inclination and of the 
 
 want of uniformity in the motion of the Node . . 118 
 
 133 TTjR construction to represent these changes . . 119 
 
 SELECTION OF EXAMINATION QUESTIONS 
 
 From College and Senate-Honse Papers and from the Moderatorship 
 
 and Fellowship Examinations at Trinity College, Dublin . 121 
 
LUNAR THEORY. 
 
 CHAPTER 1. 
 
 INTRODUCTION. 
 
 Before proceeding to the consideration of the moon's 
 motion, it will be desirable to say a few words on the law 
 of attractions, and on the peculiar circumstances which enable 
 us to simplify the present investigation. 
 
 1. The law of universal gravitation, as laid down by 
 Newton, is that " Every particle in the universe attracts every 
 other particle^ with a force varying directly as the mass of the 
 attracting particle and inversely as the square of the distance 
 between them." 
 
 The truth of this law cannot be established by abstract 
 reasoning ; but as it is found that the motions of the heavenly 
 bodies, calculated on the assumption of its truth, agree more 
 and more closely with the observed motions the more strictly 
 our calculations are performed, we have every reason to 
 consider the law as an established truth, and to attribute 
 any slight discrepancy between the results of calculation and 
 observation to instrumental errors, to an incomplete analysis, 
 or to our ignorance of the existence of some of the dis- 
 turbing causes. 
 
 Of the last source of error there is a remarkable ex- 
 ample connected with the discovery of the planet Neptune, 
 
2 LUNAR THEORY. 
 
 •which became known to us, as one of the bodies of our 
 system * solely by means of the perturbations it produced in 
 the calculated places of the planet Uranus. These pertur- 
 bations were too great to be attributed wholly to errors of 
 instruments or of calculation ; and therefore, either the law 
 of universal gravitation ceased to hold for a body so remote 
 as Uranus, or else some unknown cause was disturbing the 
 path of the planet. The first supposition was too repugnant 
 to an astronomer of the nineteenth century to be entertained 
 until all others had failed, — and the second supposition led 
 to the detection of Neptune. The distinguished names of 
 Adams and Le Verrier will be for ever connected with the 
 history of this planet, and their solution of the diflScult in- 
 verse problem : — " Given the perturbations caused by an 
 unknown planet, determine, on the assumption of the truth 
 of Newton's law, the orbit and position of this disturbing 
 body," — will always be considered one of the triumphs of 
 Mathematical Science. 
 
 Another remarkable instance of the vindication of Newton's 
 law, where it seemed at first to be at fault, will be found 
 mentioned in Chap. vi. in connection with the motion of 
 the Lunar Apogee. The error was owing to an incomplete 
 analysis; but, when the calculations were carried out more 
 fully, the strongest confirmation of the law was afforded by 
 the exact agreement of the results with observation. 
 
 We shall proceed to apply the law of gravitation to in- 
 vestigate the circumstances of the moon's motion ; and then 
 see how it will enable us to assign her position at any time 
 when sufficient data have been obtained by observation. 
 
 2. The problem would be one of extreme, if not insur- 
 mountable difficulty, if we had to take into account simul- 
 
 * It had been seen by Dr. Laniont at Munich, one year before its being known 
 to be a planet. '• Solar System, by J. 11. Hind." 
 
INTRODUCTION. 3 
 
 taneonsly the actions of the earth, sun, planets, &c., on the 
 moon ; but fortunately the planets are so small or so distant 
 that their action may be neglected — at any rate to the order 
 of our present approximation — and the attraction of the 
 earth, on account of its proximity, is very much greater 
 than the disturbing* action of the sun notwithstanding bis 
 enormous mass. We may therefore treat the question as 
 that of one body, the moon, revolving about another body, 
 the earth, and continually disturbed by a third body, the 
 sun. This is the celebrated problem of Three Bodies. 
 
 Stated in this general form its exact solution has hitherto 
 defied the powers of the analyst. The disturbing action 
 of the sun can be expressed without diflSculty in terms of 
 the masses and distances, but the integration of the diffe- 
 rential equations cannot be effected. If however, instead 
 of retaining the whole disturbing force of the sun, which, 
 as we have said, is small compared with the earth's action, 
 we expand its expression in a series and neglect all very 
 small terms, it is then found possible to obtain a solution. 
 
 This breaking up of the expression for the sun's action is 
 obviously equivalent to a breaking up of the sun's force; 
 and if we afterwards wish to take into account the very 
 small forces which have been thus neglected, or the small 
 disturbances due to the action of the planets, the following 
 principle will shew that we may do so by considering their 
 effects separately, — and that the algebraical sum of all the 
 disturbances so obtained will, to an order of approximation 
 far beyond that which we contemplate, be the same as the 
 disturbance due to their simultaneous action. We shall find 
 
 * Since the snn attracts both the earth and moon, it is clear that its effects on 
 the moon's motion relatively to the earth, or the disturbing force, will not be the 
 same as the absolute force on either body. The absolute force of the sun on 
 the moon is more than double that of the earth (Art. 99), but the relative force 
 does not exceed j^j of the earth's action (Art. 25). 
 
4 LUNAR THEORY. 
 
 an application of this principle in the investigation of the 
 parallactic inequality. 
 
 Principle of Superposition of Small Disturbances. 
 
 3. Let a particle be moving under the action of any 
 
 number of independent forces, 
 
 some of which are very small, 
 
 and let A be the position of the 
 
 particle at any instant. Let two of these small forces w,, m^ 
 
 be omitted, and suppose the path of the particle under the 
 
 action of the remaining forces to be AP in any given time. 
 Let AP^ be the path which would have been described 
 
 in the same time if »i, also had acted; AP^ diflfering very 
 
 slightly from AP^ the disturbance being PP^. 
 
 Similarly, if m^ instead of m, had acted, suppose AP^ to 
 
 represent the disturbed path, PP^ being the disturbance, 
 [AP, AP^, AP^ are not necessarily in the same plane, nor 
 
 even plane curves). 
 
 Lastly, let AQ be the actual path of the body when both 
 
 »n, and to, act. Join P, Q. 
 
 The two disturbances PP^ and P^Q, being due to the 
 action of the same small force m^ on the slightly diflferent 
 paths AP and AP^, must be very nearly equal both in 
 magnitude and in direction. What difference exists between 
 them must be a very small fraction of either disturbance, 
 and may be neglected compared with the original path. 
 We may, in fact, expect that this difference which is the 
 disturbance due to wi, of a disturbance due to m^ will be 
 of an order compounded of the orders of the two. Thus, 
 if the disturbance due to m^ and m^ be respectively of the 
 second and fourth orders, the difference between P^Q and 
 PP^ would probably be of the sixth order compared with 
 A P. Therefore P^Q may be considered parallel and equal 
 to PP,. 
 
SUPERPOSITION OF SMALL DISTURBANCES. 5 
 
 Hence the projection of the whole disturbance PQ on any 
 straight line, being equal to the algebraical sum of the pro- 
 jections of PP^ and P^Q, will be equal to the algebraical sum 
 of the projections of the separate disturbances PP^, PP^. 
 
 Next, let there be three small disturbing forces m,, wi,, m^. 
 We may consider the joint action of the two w^, w^, as one 
 small disturbing force ; therefore, by what precedes, the total 
 disturbance along any axis will be the sum of the separate 
 disturbances of jn, and of the system Wj, »«,; but this last is 
 the sum of the separate disturbances of m,^ and m, : therefore 
 the whole disturbance equals the sum of the three separate 
 disturbances. 
 
 4. This reasoning can evidently be extended to any 
 number of forces ; and if ar, y, a be the coordinates of the 
 disturbed particle, ^ (a;, ^, z) any function of x, y, z\ the 
 disturbance produced in 4> (a^j ^> 2) will be 
 
 S^ (a;, y, a) = -j^ Sa; 4 — h + 'T' ^^> omitting {Sx)', &c., 
 
 where Sx=Bxj + Sx^+...= sum of disturbances along axis of x 
 
 due to separate forces, 
 
 Si/=Si/^-^By^-i-...= along axis of jr, 
 
 Sa=Sa, +Sz,+...= along axis of a; 
 
 therefore B<j> [x, 3', «) = ^ ^^. + ^ ^^1 + ^ ^^1 
 
 = S,<t> {x, y, z) 4 B^4> [X, y, z) + &c., 
 or total disturbance equals sum of separate disturbances, 
 which establishes the principle. 
 
 Since ^ (x, y, z) may be the radius vector, the latitude 
 or the longitude of the disturbed body, it follows that the 
 
LUNAR THEORY. 
 
 total disturbance in any of these elements is the sum of the 
 partial disturbances. 
 
 5. But we must still prove another proposition, without 
 which the problem, though so far simplified, would scarcely 
 be less complicated than in its most general form. 
 
 Newton's law refers to particles, whereas the sun, earth, 
 and moon are large nearly spherical bodies, and it becomes 
 necessary to examine the mutual action of such bodies. 
 Now, it happens that, with this law of force, the attraction 
 of one sphere on another can be correctly obtained, and 
 leaves the question in exactly the same state as if they were 
 particles. [Princip. lib. I. prop. 75.) 
 
 Attractions of Spherical Bodies. 
 
 6. Let P be a particle situated at a distance OP=a from 
 the centre of a sphere of uniform 
 density p and radius c. The 
 particle being without the sphere, 
 a>c. -E- 
 
 Let the whole sphere be divi 
 ded into circular laminae by planes 
 perpendicular to OP. Let SQ be 
 one of these. PS=x, PQ = z, and thickness of lamina =Sx. 
 
 Next, let this lamina be divided into concentric rings. 
 Let RS = r be the radius of one of these rings and Br its 
 breadth, ^ BPO = 6 ; 
 therefore r = x tan ^, 
 
 Br = x sec'd.Sff. 
 
 The attraction of an element B of this ring on the particle 
 
 mass of element , __, , , , , 
 
 - along PM, and the resolved part 
 
 P will be 
 
 PR' 
 
 of this along PO will be 
 
 mass of element 
 x' sec" 6* 
 
 cos^. 
 
ATTKACTIONS 
 
 But the resultant attraction of the whole ring will clearly 
 be the sum of the resolved parts along PO of the attractions 
 of its constituent elements ; therefore, 
 
 attraction of ring = — 4-^ — j^ cos 6 = lira sin 6hx . Sd ; 
 »" sec^ 
 
 rcos-i- 
 
 therefore, attraction of whole lamina SQ=2TTpSxj sm O.dO 
 
 Jo 
 
 = 2'n-pSx(l--). 
 
 Again, a' = as' + c' - (a — x)' = 2ax - (a' - c') ; 
 therefore sSz = aSx, 
 
 9~2 ^^ ) 5 
 
 r 1^ , ,. „ f«' 2" (a'-c')2) 
 
 .•. attraction of whole sphere = 27rp -^ ;^ r-; — ^ — ;, ^-^f 
 
 ^ '^ (2a 6a' 2a' j 
 
 (from 2 = a — c to z = a-\-c) 
 
 '^ 3a' 
 1/ 
 
 47roc 
 where Jf = mass of sphere = — — . 
 
 o 
 
 Hence, the attraction of the whole sphere is precisely the 
 same as if the whole mass were condensed into its centre. 
 
 COE. 1. The attraction of a shell (radius c and thickness 
 So) will be obtained from the preceding expression by diflfe- 
 rentiating it with respect to c, and is 
 
 „ , ,, 47rpc'. 8c mass of shell 
 
 attraction ot shell = — —= — = ^ , 
 
 a a 
 
 the same as if the mass were collected at its centre. 
 
8 LUNAR THEORY. 
 
 Cor. 2. Therefore, the attraction of a heterogeneous sphere 
 on an external particle will be the same as if the whole mass 
 were condensed into its centre, provided the density he the same 
 at all points equally distant from the centre, for then the 
 whole sphere may be considered as the aggregate of an 
 infinite number of uniform shells. 
 
 7. Let us now consider the case of one sphere attract- 
 ing another. Suppose P in the preceding article to be an 
 elementary particle of a sphere M', whose centre 0' suppose 
 at a distance a from 0. Then, since action and reaction 
 are equal and opposite, P will attract the whole sphere M 
 just as it would do a particle of mass M placed at 0. The 
 same is true of all the elementary particles which compose 
 the sphere M\ therefore the sphere M' will attract the sphere 
 M as if the whole mass of the latter were condensed into its 
 centre ; but the attraction of the sphere M' on a particle 
 is the same as if the attracting sphere were condensed into 
 its centre 0' ; therefore, 
 
 Two spheres attract one another as if the whole matter of 
 each sphere were collected at its centre. 
 
 8. This remarkable result, which, as may be shewn, holds 
 only when the law of attraction is that of the Inverse square 
 of the distance, or that of the direct distance, or a com- 
 bination of these by addition or subtraction, reduces the 
 problem of the sun, earth, and moon to that of three par- 
 ticles. The slight error due to the bodies not being perfect 
 spheres is here neglected, being of an order higher than 
 that to which we intend to carry the present investiga- 
 tion : this error, however, though very small, is appreciable, 
 and when a nearer approximation is required, it becomes 
 necessary to have regard to this circumstance. (See Ap- 
 pendix, Art. 109). 
 
( 9 ) 
 
 CHAPTER 11. 
 
 MOTION EELATIVE TO THE EAETH. 
 
 9, When a number of particles are in motion under their 
 mutual attractions or other forces, and the motion relatively 
 to one of them is required, we must bring that one to rest 
 and then keep it at rest, without altering the relative motions 
 of the others with respect to it. 
 
 Now, first, the chosen particle will be brought to rest 
 by giving it at any instant a velocity equal and opposite to 
 .that which it has at that instant; secondly, it will be kept 
 at rest by applying to it accelerating forces equal and 
 opposite to those which act upon it. 
 
 Therefore give the same velocity and apply the same 
 accelerating forces to all the bodies of the system, and their 
 absolute motions about the chosen body, which is now at rest, 
 will be the same as their relative motions previously. 
 
 Problem of Two Bodies. 
 
 10. As the sun disturbs the moon's motion with respect 
 to the earth, it is important to know what that motion would 
 have been if no disturbance had existed, or generally : — 
 
 Two bodies attracting one another with forces varying 
 directly as the mass and inversely as the square of the dis- 
 tance, to determine the orbit of one relatively to the other. 
 
 Let M, M' be the masses of the bodies, M' being the body 
 whose motion relatively \.o M \s required, r the distance 
 
 c 
 
10 LUNAR THEORY. 
 
 between them at any time t, and 6 the angle between r and 
 
 some fixed prime radius. 
 
 M 
 The accelerating force of J/ on M' equals -j- acting towards 
 
 M' . ... 
 
 M, while that of M' on M equals -^ in the opposite direction. 
 
 Therefore, by the principle above stated, we must apply to 
 
 both M and M' accelerating forces equal and opposite to this 
 
 latter force, and M' will move about M fixed, the accelerating 
 
 M+M' 1 
 
 force on M' being j — = /j,u', if fi = M+M' and r = - . 
 
 Hence, 
 
 lere h = r 
 and integrating 
 
 where h = r' -;-= twice the area described in a unit of time, 
 
 M = Ti{l +ecos(^ -«)}i 
 
 e and a being constants to be determined by the circumstances 
 of the motion at any given time. 
 
 This is the equation to a conic section referred to its 
 
 focus, the eccentricity being e, the semi-latus rectum — , and 
 
 the angle made by the apse line with the prime radius a. 
 
 By observations made on the actual path of the moon, 
 which cannot diflfer very widely from its undisturbed orbit, 
 we infer that this latter would be an ellipse with an 
 eccentricity about -j^. 
 
 In the same manner, if the sun and earth be the two 
 bodies considered, the relative orbit would be an ellipse with 
 an eccentricity about ^g. 
 
 11, The angle — a between the radius vector and the 
 apse line is called the true anomaly. 
 
two BODIES. 11 
 
 If n is the angular velocity of a radius vector which, 
 moving uniformly, would accomplish its revolution in the 
 same time as the true one, both passing through the apse 
 at the same instant; then nt + s-a is called the mean 
 anomaly where e is a constant depending on the instant 
 from which the time is reckoned, its value being the angle 
 between the prime radius and the uniformly revolving one 
 when t = Q. 
 
 Thus, let MT be the fixed line or prime radius, 
 A the apse, 
 M' the moving body 
 at time t, 
 
 Mfi. the uniformly re- 
 volving radius at same time, the 
 direction of motion being repre- 
 sented by the arrow. 
 
 And, let MD be the position of M/i when f = 0, 
 then TMD = e and is called the epoch,* 
 
 DMiJ. = nt, 
 
 tMA = a = longitude of the apse ; 
 therefore, mean anomaly = AMfL = w< + s - a, 
 true anomaly = AMM' = tMM ' - TMA = 0-a. 
 
 12. To express the mean anomaly in terms of the true in 
 a series ascending according to the powers of q.^ as far as e'. 
 
 27r „ 2 area 
 
 n = ■ J. . — =29r-=- — J — 
 
 periodic time h 
 
 2-rrh h 
 
 2irab d'^/(l-e') 
 
 • We shall see (Art. 36) that the introduction of the epoch i is avoided 
 in the Lunar Theory by a particular assumption ; but in the Planetary it forma 
 one of the important elements of the orbit. 
 
12 LUNAR THEORY. 
 
 therefore h = mo' (1 — e')*, 
 
 dt r' a'(l-ey 1 
 
 dd h h ' {l+ecos(^-a)j' 
 
 = -(l-e')*{l+eco8(0-a)P 
 
 = - (1 - f e") {1 - 2e cos (^ - a) + Se" cos" [6 - a)} 
 
 = - {l-2ecos(^-o) + |e'cos2(^-a)}; 
 
 therefore n< + e = - 2e sin {6 - a) + f e' sin 2 (0 - o), 
 or (w« + e - a) = (^ - o) - 2c sin (0 - a) + f e" sin 2 (^ - a), 
 the required relation. 
 
 13. To express the true anomaly in terms of the mean to 
 the same order of approximation. 
 
 ^ - a = M« + e - a + 2e sin (0 - a) - f e" sin 2 (5 - a)...{l) ; 
 
 .*. 6 — a = nt-\-& — a first approximation. 
 
 Substituting this in the first small terms of (1), we get 
 
 5 - a = w< + e — a + 2e sin (n« + E — a)...a second approximation. 
 
 Substitute the second approximation in that small term 
 of (1) which is multiplied by e, and the first approximation 
 in that multiplied by e', the result will be correct to that 
 term, and gives 
 
 ^ - a = ni + e — a + 2e sin {n< + E — a + 2e sin (n< + e — a)} 
 
 - fe'sin2 (ni + E -a) 
 
 = n< + e - a + 2e sin (wi + s — a) 
 
 + 4c''cos(w«+£-a) 8in(nf+e— a)— fe''sin2(n<+E-a) 
 
 = nt + B-a-i-2e sin [nt + t-a) + f e" sin2 {nt + e- o), 
 
 the required relation. 
 
THKEE BODIES. 13 
 
 The development could be carried on by the same process 
 to any power of e, the coefficient of e' would be found 
 
 If sin 3 («< + e - a) - J sin (nt + t-a), 
 
 but in what follows we shall not require anything beyond e". 
 
 Problem of Three Bodies, 
 
 14. In order to fix the position of the moon with respect 
 to the centre of the earth, which, by means of the process 
 described in Art. (9), is supposed brought to rest, we must 
 have some determinate invariable plane passing through the 
 earth's centre to which the motion may be referred. 
 
 If the sun and earth were the only bodies in the universe, 
 then the plane of the elliptic orbit (in which, according 
 to the last section, the motion would take place) would be 
 fixed, and might be taken as the plane of reference ; but, 
 as soon as we take into account the action of the moon 
 and planets — especially the moon — the plane ceases to be 
 fixed, and some other plane must be found not affected by 
 these disturbances. 
 
 Theory teaches us that such a plane exists,* but as its 
 exact determination can only be the work of time, the 
 
 * See Poinsot, " Theorie et determiTiation de V equateur du systeme solaire,'^ 
 where he proves that an invariable plane exists for the solar system, that is, a 
 plane whose position relatively to the fixed stars will always be the same, 
 whatever changes the orbits of the planets may experience; but as its position 
 depends on the moments of inertia of the sun, planets, and satellites, and there- 
 fore on their internal conformation, it cannot be determined a priori, and ages 
 mnst elapse before observation can furnish sufficient data for doing so a 
 posteriori. 
 
 This result Poinsot obtains on the supposition that the solar system is a 
 free system ; but it is possible, as he furthermore remarks, nay probable, that the 
 stars exert some action upon it, it follows that this invariable plane may itself be 
 variable, though the change must, according to our ideas of time and space, 
 be indefinitely slow and small. 
 
14 LUNAR THEORY. 
 
 following theorem will supply us with a plane whose motion 
 is extremely slow, and which may for a very long period, 
 and to a degree of approximation far beyond that to which 
 we shall carry our investigations, he considered as fixed and 
 coinciding with its position at present. 
 
 In what follows the action of the planets is too small 
 to be taken into account. 
 
 15. The centre of gravity of the earth and moon descrihes 
 relatively to the sun an orbit very nearly in one plane and 
 elliptic; the square of the ratio of the distances of the mooo 
 and sun from the earth being neglected.* 
 
 Let S, E, M be the centres of the sun, earth, and moon^ 
 G the centre of gravity of the last two. Now the motion 
 of O is the same as if the whole mass ^+ilf were collected 
 there and acted on by forces equal and parallel to the moving 
 forces which act on E and M. The whole force on O is 
 therefore in the plane SEM; join SG. 
 
 Let I SGM= to, and let m be the sun's absolute force. 
 Moving force on E= „!-,, in ES, 
 
 moving force on M= ' in MS. 
 
 * This ratio is about 7^, and, as we shall see Art. (21), such a quantity 
 we shall consider as of the 2nd order of small quantities, and its square therefore 
 of the 4th order. Our investigations are carried to the 2nd order only. 
 
PLANE OF REFERENCE. 15 
 
 These applied to O parallel to themselves are equi- 
 valent to 
 
 m'.E.OE m'.M.GM. ,. . ^„ ] . , 
 
 — 'SW 'SW — "^ direction GM, by the 
 
 I trianffle 
 and -^•■E-SG m'.M.SG ^^. Lf forces. 
 
 , ^_M__ M+E 
 
 ° GM~ GE~ ME ' 
 
 whence E. GE= M. GM= {M+ E) ^^^ • 
 
 Therefore substituting, and dividing by M+ E, we get 
 accelerating forces* 
 
 , GM.GE f 1 1 N . ,. . ^,, 
 
 "* • -ME- [-SE" - SMJ "* ^"■^'^*'°° ^^' 
 
 J m'.SG ( E M \ ^„ 
 
 ^"^^ ^T^ Km + -sM") ^^• 
 
 Again, 8E' = SG*-\-GE' + 2SG.GEcosto 
 
 therefore — = -^ (\ - ^^ cos w\ 
 
 SE^ SG^ \ SG J\ omitting quantities 
 
 . ., , 1 1 /, 3GM \ to be neglected, 
 
 simdarly ^, = ^^-3 (l + -^ cos.j 
 
 * In strictness it would be necessary, since we have brought 5 to rest to 
 apply to both M and £, and therefore to G, accelerating forces equal and opposite 
 to those which E and M themselves exert on <S ; but the mass of £ is so large 
 compared with those of E and M, that we may safely neglect these forces in 
 this approximate determination of the path of G, the error being of a still 
 
 higher order than that introduced by the neglect of I -^ j . 
 
16 LUNAR THEORY. 
 
 Therefore the accelerating force in the direction GM 
 Sm'.GM.OE 
 
 o , 1 .• ^ c „, GM GE 
 
 = — 3 (accelerating torce or sun on G) -^^ . -^^ coseo 
 
 = 0, according to standard of approximation adopted. 
 
 And the accelerating force in GS 
 
 m'.SG (E + M 3{E.GE-M.GM] ) 
 
 = lUE tSG^- SG* '''i 
 
 m , . . 
 
 — 'oTri ^^ the same approximation. 
 
 Hence, the force on (? is a central force tending to S 
 and varying inversely as the square of the distance ; there- 
 fore the orbit of G about S is very approximately an ellipse 
 with 8 in the focus ; and the plane of this ellipse is, as far 
 as our investigations are concerned, a fixed plane. 
 
 This fixed plane is called the plane of the ecliptic^ or 
 simply the ecliptic. 
 
 16. We shall now suppose a plane through the earth's 
 centre parallel to the ecliptic; and this, when the earth is 
 brought to rest, will give us the fixed plane of reference 
 we require (14); the ecliptic then making small monthly 
 oscillations from one side to the other of our fixed plane. 
 
 17. Since G describes an ellipse relatively to the sun, 
 the sun will describe the same ellipse relatively to G ; but, 
 as seen from the earth, the orbit of the sun will be slightly 
 diflferent, as we shall now shew. 
 
 First: The sun will have a latitude, that is, will be out 
 of the plane of reference; and this latitude will be of the 
 same name (north or south) as that of the moon, and 
 deducible from it when the ratio of the distances of the two 
 
EREORS IN sun's POSITION. 17 
 
 bodies from the earth, and that the masses of the earth 
 
 and moon are known. For if S'EM' be the fixed plane 
 through E^ and <S', G\ M' the projections of 8, G, M, then 
 SES' is the sun's latitude, and MEM' is the moon's; and 
 the two bodies are obviously both on the same side of 
 the plane. 
 
 . , , , ,. 88' GG' £<?. sin (moon's lat.) 
 sm(8un 8 lat.) = ^ = -^ ^ 
 
 M EM . , , , , 
 = -r, T,i . ttjt . sm (moon s lat). 
 
 E+M ES ^ ' 
 
 Now, it is known that M is about g'^ of E, 
 
 and EM ^^ of ES; 
 
 ^i r • / 5 1 i. N sin (moon's lat.) , 
 
 therefore smlsun s lat.) = — ^^-i:^:^^^ nearly. 
 
 32400 "^ 
 
 And as the moon's latitude never exceeds 5° 9', the sun's 
 latitude will always be less than 1". 
 
 Secondly, as to the sun's longitude: let ET be the 
 direction of the first point of Aries, — that is, a fixed line 
 in our plane of reference from which the longitudes of the 
 bodies are reckoned. Then TE8' = 6' is the sun's longitude 
 as seen from E; but, as seen from G, the sun's longitude 
 would be the angle between G8 and a line through G (not 
 drawn in the figure) parallel to ET. 
 
 The difference between these two longitudes is the angle 
 E8'G'; a.nA,i{EG'8' = <o, 
 
 . T,„, ™ EG' . sina) . ^ , 
 
 smE8 G =-g^ ®'"*"~ 32400 approximately; 
 
18 LUNAR THEORY. 
 
 therefore s\n ES'G' never exceeds ^^^^^, and ES'G' is a 
 small angle not exceeding 7". 
 
 Also ES' ~ 5'<?' < EG' < 5^i^^ S' G'. 
 
 18. Now, by assuming the longitude and distance of the 
 sun as seen from E to be the same as -when seen from (?, 
 we commit the above small errors in the position of jS; 
 that is, we assume the sun to be at S" instead of 5, S' S" 
 being drawn equal and parallel to G'E. If our object were 
 the determination of the sun's position, it would be neces- 
 sary to take this into account ; but when investigating the 
 motion of the moon, the disturbing action of the sun will 
 not, on account of his great distance, be appreciably altered 
 by supposing him to be at S" instead of S; and the errors 
 thus introduced are much too small to be considered. 
 
 Hence we may assume that the motion of the sun about 
 the earth at rest is an ellipse having the earth for its focus, 
 and its equation 
 
 u=a{l+e' cos(e'-f)}, 
 
 where f is the longitude of the sun's perigee and e' = j^j^ is 
 the eccentricity, and we are safe that no appreciable error 
 will ensue in the determination of the moon's place.* 
 
 * That is, as far as the three bodies alone are concerned; — but, since the 
 attractions of the planets may, and in fact do, disturb the eUiptic orbit of 
 the sun about G, the same cause will disturb the assumed orbit about H. A 
 remarkable result of this disturbance is noticed in Appendix, Art. (108). 
 
( 19 ) 
 
 CHAPTER III. 
 
 HIGOEOUS DIFFERENTIAL EQUATIONS OF THE MOON'S MOTION 
 ANB APPBOXTMATE EXPBESSIONS OF THE FORCES. 
 
 19. The earth having been reduced to rest by the process 
 
 described in Art. (9), let us take its ^1 
 
 centre S for origin. Let r, 6 be the ^~~^^^^ 
 
 polar coordinates of the projection M' "^^^^^^^ 
 
 of the moon on the fixed plane of re- 
 ference, d being the longitude TEM' measured from a fixed 
 line EX in that plane. Let s be the tangent of the moon's 
 latitude MEM\ so that MM' = rs. 
 
 Next, let the accelerating forces which act Upon the moon 
 be resolved into these three : 
 
 P parallel to the projected radius M'E and towards the earth, 
 T parallel to the fixed plane, perpendicular to P, and in the 
 
 direction of 6 increasing, 
 8 perpendicular to the fixed plane and towards it. 
 
 The equations of motion will be 
 
 lU"^-^ «. 
 
 5-Kf)'=-^- ("'. 
 
 •^—s (iii). 
 
 20. These three equations for determining the moon's 
 motion take the time t for independent variable ; but it will 
 
20 LUNAR THEORY. 
 
 be more convenient in the following process to consider the 
 longitude as such, and our next step will be to change the 
 independent variable from t to 6. 
 From (i) we get 
 
 ,d0 d ( ,dd\_^,de 
 
 dt) 
 
 dt dt \ dt) dt 
 
 therefore [r' ^) = A' + 2 iTt'dd, 
 
 = H' suppose, whence B.-t^= Tr^. 
 
 Therefore — - = -5 = Hu', if m = - , 
 
 at r r 
 
 dt 1 1 /, „ r ^ ja\'* 
 
 de = -m = M\t"'^mn ' 
 
 an important equation connecting the time t with the longi- 
 tude 6. 
 
 . . dr dr dO -ndu 
 
 ^^^'""^Jt^Je-Tt^-^Te^ 
 
 d^r d I Tjdu\ -jy J d /—.cZmN 
 
 5? = dt r^^J =- ^" Te K^Je) • 
 
 Substitute in (ii), 
 
 therefore if.- ^^(i7|) + £-.•.?, 
 
 ''■''■(S-)+»-K-^ <^)^ 
 
 or 
 
 and substituting for H' and for S-^ their values A'+ 2 /-^ d9 
 
 T 
 and -^ , then dividing by A'm', and transposing, 
 
 t?'M , P T du ^ (d'u \ [ T J. 
 
 5e^ + " = AV- A^' d0-^[w + V JAV'^^5 
 
 this is called the differential equation of the moon's radius 
 vector. 
 
DIFFERENTIAL EQUATIONS. 21 
 
 ( ds du ] 
 
 dydd'^cw de\ d 
 
 H 
 
 ( ds du\) 
 
 [''dd-'do)\ 
 
 dt^ m' ■ dti dt 
 
 „j , / d^'s d^u\ -ndll , / ds du\ 
 
 ^^'''Vdb^-'w)+^Te^[''de-'dep 
 from (A P. = w(.g+.s) + 4f.42; 
 
 therefore P.- ^= W (g + «) + ^f uf^; 
 
 therefore 
 
 ^ _ Ps-S T ds (d's \ [T ,. 
 
 this is the differential equation of the moon's latitude. 
 
 21. If these three equations could be integrated under 
 these general forms, the problem of the moon's motion would 
 be completely solved ; for as only four variables m, ^, «, and 
 t are involved (the accelerating forces P, T, and 8 being 
 functions of these), the values of three of them, as m, 6, s, 
 could be obtained in terms of the fourth t; that is, the radius 
 vector, longitude, and latitude would be known correspond- 
 ing to a given time. 
 
 But the integration has never yet been efiFected, except 
 for particular values of P, T, and S ; and the method which 
 we are in consequence forced to adopt is that of successive 
 approximation, by which the values of u, 6, and s are 
 obtained in series in which the terms proceed according to 
 ascending powers of certain small fractions. Some one 
 fraction being chosen as a standard with which all other 
 are compared, the order of the approximation is esteemed 
 by the highest power of the small fractions retained. 
 
22 LUNAR THEORY. 
 
 It is usual to consider ^^^ as a small fraction of the first order, 
 consequently ^ of ^■s = zhis '^ second 
 
 W^riyis third 
 
 and so on, other fractions being considered as of the 1st, 
 2nd, &c. orders, according as they more nearly coincide with 
 
 22. It is necessary therefore, before we can approximate 
 at all, that we should have a previous knowledge (a rough 
 one is sufficient) of the values of some of the quantities 
 involved in our investigations; and for this knowledge we 
 must have recourse to observation. 
 
 We shall therefore assume as data the following resnlta 
 of observation : 
 
 (1) The moon moves in longitude about thirteen times 
 as fast as the sun. The ratio of the mean motions in longi- 
 tude represented by m is therefore about ^5, and may be 
 considered as of the 1st order.* 
 
 (2) The sun's distance from the earth is about 400 times 
 as great as the moon's distance. 
 
 Hence the ratio of the mean distances = -f^jf is of the 
 second order.f 
 
 (3) The eccentricity e' of the elliptic orbit which the 
 sun approximately describes about the earth is about 5^, 
 and this, approaching nearer in value to ^ than to -jj-g, 
 will be considered as of the 1st order. 
 
 (4) During one revolution, the moon moves pretty ac- 
 curately in a plane Inclined to the plane of the ecliptic at 
 
 • This appiozimate Talne of m is easily obtained ; — the moon is foond ta 
 perform the tour of the heavens, letvuning to the same position among the 
 fixed stars, in about 27^ days; the son takes 365^ days to accomplish the 
 same journey. 
 
 ■)• The distances of the luminaries may be calculated from their horizontal 
 parallaxes, found by observations made at remote geographical stations. (See 
 the Author's Astrotiomy, chap. XVI.), 
 
EXPRESSIONS OP THE PORCES. 23 
 
 an angle whose tangent is about ^, and therefore of the 
 1st order.* 
 
 (5) Its orbit in this plane is very nearly an ellipse having 
 the centre of the earth in its focus, and whose eccentricity is 
 about equal to our standard of small fractions of the 1st order, 
 ^*^* sV j ^^^ tl'is ^ill also be very nearly true of the projec- 
 tion of the orbit on the plane of the ecliptic.f 
 
 To calculate the values of the forces. 
 
 23. We are now in possession of the data requisite for 
 beginning our approximations, and we shall proceed to the 
 determination of the values of P, T, and S in terms of the 
 coordinates of the positions of the sun and moon. 
 
 Let iS, E^ M be the centres of the sun, earth, and moon, 
 
 »«', ^, M their masses, 
 
 E', M\ the projections on the plane of the ecliptic, 
 
 G the centre of gravity of E and M. 
 
 * That the moon's orbit during one revolution is very nearly a plane inclined 
 as we have stated, will be found by noting her position day after day among the 
 fixed stars; and the sun's path having previously been ascertained in a similar 
 way, the rules of Spherical Trigonometry will easily enable us to verify these 
 statements. (Astronomy, p, 92). 
 
 t The elliptic nature and the value of the eccentricity of the moon's orbit may 
 be found by daily observation of her parallax, whence her distance from the earth's 
 centre may be determined : corresponding observations of her place in the heavens 
 being taken, and corrected for parallax to reduce them to the earth's centre, will 
 determine her angular motion. Lines proportional to the distances being then 
 drawn from a point in the proper directions, the extremities mark out the form 
 of the moon's orbit, 
 
 A similar method applied to observations of the diameter of the sun will 
 determine the eccentricity of its orbit. (Astronovit/, p. 138). 
 
24 
 
 LUNAR THEORY. 
 
 The forces we have to take into account are, the forces 
 ■which act directly on M, and forces equal and opposite to 
 those whicli act on E; — these last being applied to the whole 
 system so that -E may be a fixed point. 
 
 Attraction of E= 
 attraction of S 
 mMS 
 
 Forces on M. 
 E 
 
 m 
 ^ SM' 
 equivalent to 
 
 ME'" 
 
 rn.MG 
 
 .in ME 
 
 .ia ME 
 
 SM' 
 
 m'.SG , _„ 
 -g^^par'.toG'^ 
 
 Forces on E. 
 
 M 
 
 Attraction of M= ^^r, in EM 
 
 ME' 
 
 attraction of 8 
 = ^^.inES 
 equivalent to 
 
 ^'.EG 
 
 SE' 
 vi'.SG 
 
 .in EM 
 
 SE' 
 
 pari, to G8 
 
 Therefore, the whole attraction upon M, when E is 
 brought to rest, is 
 
 E+M ,/MG EG\. ,,„ 
 
 and m'SG f -^yj? - -qwz ) parallel to G8. 
 
 These are the rigorous expressions of the accelerating 
 forces on M, and they can be expressed in terms of the 
 masses and coordinates of the bodies. 
 
 Approximate values of P, J", S. 
 24. Our investigations will be carried only to the second 
 order ; it will be sufficient therefore if, in the preceding, we 
 neglect small quantities of the fourth and higher orders. 
 Let^ = ^+i/, 
 
 MGM' = tan~'s = moon's latitude, 
 
 ^'T the direction of the first point of Aries, 
 
 S(? = r' = -i, ; irE'S= & = longitude of sun, 
 
 M'E' = r = - ; L^E'M' = = longitude of moon, 
 .•. SE'M' = 0-6' = difiference of longitude of sun and moon. 
 
EXPRESSIONS OP THE FORCES. 25 
 
 Now, SAP = SM''+ MM" 
 
 = SG'+ GM''-2SG.OM'co3SGM'-^MM'* 
 
 „/, ^GM' „^,„ GM\ 
 = /'(!- 2 — —coBSGM' + —f^y, 
 
 therefore ^^, = 7» V + ~^ °°^l^ - ^ Jj 5 
 
 for e-ff or )Si;'J(f' diflfers from SGM' by less than 7', 
 
 Art. (17), and ( — p-J is neglected, being of the fourth order. 
 
 Similarly, -^^ = -^ ]l ^,— cos(0 - ^')[ 5 therefore 
 
 the accelerating forces on the moon are approximately 
 
 -r^ + -', {MG +GE) in direction ME, 
 
 and ^(<?Jf'+ GE') co8(0- &) parallel to G8; 
 
 whence P= f-f^ + ^' M^] cosMGM'- ^ M'E' cos'l^-^') 
 \AJE r 1 r 
 
 :i+^-^;r» {l + cos2{5-0)J 
 
 r^(l+/)* "^ 2r 
 
 T= - ^ M'E' cos {e - &) sin [6 - ff) 
 
 = _3^'sln2(0-5'), 
 
 S={^. + %MEynMGM 
 
 ( fi m'r \/(l 4 s') ) s 
 
 ' '3 
 
 ' '3 
 
 P,_5=-^^{| + |cos2(5-^')}. 
 
26 LUNAR THEORY. 
 
 The differential equations in Art. (20), when these values 
 of the forces are substituted in them, would contain the 
 variables u and 6'; but u' is given in terms of 6' by the 
 equation of Art. (18) u' = a'\l +e' cos(0' — f)}, and we shall 
 find means to establish a connexion between and 6\ which 
 win enable us to eliminate the latter. 
 
 25. Before proceeding further, it will be important to 
 consider the order of the disturbing efiFect of the sun's action, 
 compared with the direct action of the earth. If we examine 
 the values of P, T, and S, it will be found that the most 
 important of the terms containing m', which are clearly the 
 disturbing forces since they depend upon the sun, are in- 
 
 7/1 T 
 
 volved in the form —^ , while those independent of the sun's 
 
 action enter in the form -; . 
 r 
 
 We must therefore find the order of —^ compared with ^ , 
 
 or of-r. 
 
 Now the orbits being nearly circular, and m the ratio 
 of the mean motions. Art. (22), we have 
 
 mean motion of sun periodic time of moon 
 
 »n = -. 2i = ^ — =— T^ — -■ ;; 
 
 mean motion ot moon periodic time of sun 
 
 2m£ 27rr'^ _ 
 
 therefore —.•.—,:•. m} : 1, 
 
 or the disturbing force of the sun is of the second order 
 when compared with the direct action of the earth. 
 
( ^7 ) 
 
 CHAPTER IV. 
 
 INTEGBATION OF THE DIFFERENTIAL EQUATIONS. 
 
 Section I. 
 
 General process described. 
 
 26. If in the differential equations we substitute the 
 values of P, 1\ and S just obtained, we are led to the 
 following forms: 
 
 ^,+u = F^{u,0,s)^ 
 2^i + s=F^{u,e,s), 
 
 J^^=F,[u,0,s), 
 
 which cannot be integrated ; but it is found possible to obtain 
 a solution which will be a first approximation, if we retain 
 only the important terms on the right-hand side of each 
 equation. This first approximation being substituted in the 
 equations, the terms next in order of importance may be 
 retained and a new solution becomes possible, which in its 
 turn is the basis of a further approximation, and so on. 
 
 First, neglect the disturbing force of the sun which !s 
 of the second order, and also the moon's latitude, which, 
 as will be seen by referring to the expressions for the 
 forces (Art. 24), will either enter to the second power or else 
 in combination with the disturbing force. 
 
28 LUNAR THEORY. 
 
 When this is done the equations become integrable, and 
 values of u and s may be obtained in terms of 9 correct to 
 the first order; this value of u will then enable us to get 
 the connexion between and t to the same order. 
 
 [Let us, however, bear iu mind that the equations thus integrated are not 
 the differential equations of the moon's motions, but only approximate forma 
 of them; and it is, therefore, possible that the results obtained may be ap- 
 proximations to the true solutions only for a limited range of values- 
 
 "Whether they are so or not, will appear by comparing them with what we 
 already know of the motion &om observation ; and this previous knowledge, 
 in the event of their not being approximations, will probably suggest such 
 modifications of them as will render them so.] 
 
 To proceed to the next approximation: — We must sim- 
 plify the right-hand members of our general equations by 
 employing the values of u and s just obtained, retaining 
 now terms of the second order which we had previously 
 neglected. 
 
 The integration can always be performed when the right- 
 hand members of the equations are circular functions (sines 
 or cosines) of 6 or of its multiples; and as our first approxi- 
 mation will give us the values of u and s in that form, the 
 new expressions for the forces will also be of the same 
 character, and the equations be again integrable. 
 
 Thus we shall obtain new values of m, », t correct to the 
 second order. These values, introduced in the same manner 
 in the second members and terms of the next higher order 
 retained, will lead to a third approximation, and so on, to 
 any order ; except that if we wish to carry it on beyond 
 the third, the approximate values of the forces, given in 
 Art. (24), would no longer be sufficiently exact, and we 
 must obtain more correct values from the expressions of 
 Art. (23). 
 
 27. W^e must here stop to notice a peculiarity in these 
 equations, when solved by this process. We have said that 
 
GENERAL PROCESS OF INTEGRATION. 29 
 
 to obtain the values to any order, all terms up to that order 
 must be retained in the second members ; but it may happen 
 that a term of an order beyond that to which we are work- 
 ing would, if retained, be so altered by the integration as 
 to come within the proposed order. 
 
 Such terms must therefore not be rejected, and we 
 shall proceed to examine by what means they may be 
 recognised. 
 
 28. Suppose then that after an approximation to a certain 
 order, the substitutions for the next steps have brought the 
 equation in u to the form 
 
 ^^ + M= + Gcos(j)d+H)+ , 
 
 where the coefficient G is one order beyond that which we 
 
 intend to retain. The solution of this equation will be of 
 
 the form 
 
 M= + G' cos[p0-\-H)+ 
 
 G' being a constant which may be determined by substi- 
 tuting this value of u in the differential equation, 
 
 whence G' = ^ ; 
 
 from which we learn that if ^ diflPers very little from 1, G' 
 will be at least one order lower than G, and will come 
 within our proposed approximation ; and consequently the 
 term G coa(pd+E) must be retained in the differential 
 equation. 
 
 The equation of the moon's latitude being of the same 
 form as that of the radius vector, the same remarks apply 
 to it. 
 
 29. If p is small, G' = G very nearly, and the term 
 G cos {p6 + H] will not rise in importance in the value of u. 
 
30 LUNAR THEORY. 
 
 We should therefore be led to reject it in the differential 
 equation, except for the following reason : — 
 
 In finding the connexion between the longitude and the 
 time (one of the principal objects of the Theory), we must 
 use the equation 
 
 Now, substituting for u its value in terms of 5, we shall 
 find that such a term as G' cos{p0 -i- H) in u gives rise to 
 a term G"cos[j>0 + H) in this equation, where G' and G' 
 are of the same order, then 
 
 J= G"co8(p5 + 5), 
 
 C" 
 whence t= — aia {p6+ H); 
 
 G" 
 therefore, when p is a small quantity of the first order, — 
 
 will be one order lower than G", and the term will have 
 
 risen in importance by the integration. 
 
 T 
 But yet further, if such terms occur in =t-% , they will be 
 
 twice increased in value ; for they increase once in forming 
 
 I jj—, d6, and once again, as above, in finding t. 
 
 30. We have, therefore, the following rule : — 
 
 In approximating to any given order to the values of u 
 and s, we must, in their differential equations, retain periodical 
 terms ONE ORDER beyond the proposed one, when the coefficient 
 of 6 in their argument* is near unity. 
 
 • The angle of a periodical term is its only Tariable part and is called 
 the nrg^mieni. 
 
TEEMS TO BE RETAINED. 31 
 
 In approximating to the value of «, we must, in the same 
 
 equations, also retain terms ONE OBDEE beyond the proposed 
 
 one when the coefficient of the argument is near zero; and 
 
 T 
 when such terms occur in rf~, they must be retained TWO 
 
 OEDEEa beyond the proposed approximation* 
 
 31. We shall here, for convenience of reference, bring 
 together the equations and the approximate expressions for 
 the forces : — 
 
 <^M , P T du /d'u \ rT 
 
 5^+"= ¥Z' -l^'-de-Hde' + VJhw'^^-^^^' 
 
 4!f J. -^-1 T ds ^(d's \ rT j„ 
 dd'^'- -AV- ~ AV • 5^ - HJ^' "^ V JAV ^^ - ^^^' 
 
 T0=W['+'jh^u^^^) (^^)' 
 
 * Instead of the forces which really act on the moon, we originally substituted 
 three equivalent ones, P, T, S; these again are, by the preceding expressions, 
 
 p 
 replaced by a set of others. For, we may conceive each of the terms in j^ , &c., 
 
 to correspond to a force, — a component of P, T, oi S; each force having the 
 same argument as the term to which it corresponds, and therefore going through 
 its cycle of values in the same time. Now, by Art. (29), when the coefBcient of 
 e in the argument is near unity, the term becomes important in the radius vector, 
 and when near zero, in the longitude : hence, a force whose period is nearly the 
 same as that of the moon, produces important effects in the radius vector ; and a 
 force whose period is very long will be important in its effect on the longitude. 
 See Airy's Tracts, Planetary Theory, p. 78. 
 
32 LUNAR THEORY. 
 
 Section II. 
 To solve the Equations to the first order. 
 
 32. Neglect all temis depending on the disturbing force, 
 t'.e. those which contain m'; such terms being of the second 
 order. Art. (25). 
 
 The latitude s of the moon can never exceed the inclina- 
 tion of the orbit to the ecliptic ; but this inclination is of the 
 first order, Art. (22), therefore s is at least of the first order 
 and s' may be neglected. 
 
 W hence -> = M ; -j = : 5— = 0, 
 
 and the diflTerential equations become 
 
 d'u 
 
 y, +M 
 
 M 
 
 > 1 
 
 dO' ' H 
 
 d's 
 d'ff''^' = '-^ 
 
 whence " = p {1 + fi cos(^ — o)j ; or, writing a for ^ , 
 
 u = a {l+ecos(^-a)} [U,), 
 
 and s = ka\a[0 — 'Y) (5,), 
 
 e, a, k, 7 being the four constants introduced by integration. 
 
 33. These results are in perfect agreement with what 
 rough observations had already taught us concerning the 
 moon's motion Art. (22); for M = a {1 +e cos(^— a)} repre- 
 sents motion in an ellipse about the earth as focus. 
 
 And s = Asin(0 — 7) indicates motion in a plane inclined 
 to the ecliptic at an angle tan"'A. 
 
 For, if TOM' be the ecliptic, 
 M the moon's place, MM' an arc 
 perpendicular to the ecliptic, then 
 
SOLUTION TO FIKST OUDER. 33 
 
 Ti¥' = 5; and if TO be taken equal to 7, and OJlf joined 
 by an arc of great circle, we have 
 
 tan JOf' = ta.n MOM' sin OM' ; 
 or s = tSLuMOM' siD(0- y), 
 
 which, compared with the equation above, shews that 
 
 MOM' = t&h-'k. 
 Therefore the moon moves in a plane passing through 
 a fixed point and making a constant angle with the 
 ecliptic. 
 
 The values of a and 7 introduced in the above solutions 
 are respectively the longtitude of the apse and of the node, 
 
 34. What the equations cannot teach us, however, is 
 the magnitude of the quantities e and k. For this we must 
 have recourse to observation, and by referring to Art. (22), 
 we see that e is about -^-^ and k about ^^j, that is, both 
 quantities are of the first order. Their exact values cannot 
 yet be obtained: the means of doing so from multiplied 
 observations will be indicated further on. 
 
 35. Lastly, to find the connexion between t and 0, the 
 equation ( C) becomes, T= 0, 
 
 ^_ J^ _ J^ ] 
 
 dd ~ hi? ~ ha' [l + e cos(d/ - a)}' ' 
 
 How this is the very same equation that we had found 
 connecting t and d in the problem of two bodies, Art. (12), as 
 we ought to expect, since we have neglected the sun's action. 
 Therefore, if p be the moon's mean angular velocity, we 
 should, following the same process as in the article referred 
 to, arrive at the result 
 
 =pt+ & + 2e BiD(p + 1- a) + ^e' ain2(pt-\- e- a)+ , 
 
 which is correct only to the first order, since we have 
 
 F 
 
34 LUNAR THEORY. 
 
 rejected some terms of the second order by neglecting tlie 
 disturbing force. 
 
 36. The arbitrary constant e, introduced in the process 
 of integration, can be got rid of by a proper assumption : 
 this assumption is, that the time t is reckoned from the 
 instant when the mean value of is zero.* 
 
 For, since the mean valne of 0, found by rejecting the 
 periodical terms, is pt + e; if, when this vanishes, < = 0, we 
 must have e = ; therefore 
 
 6=pt + 2eam(ji>t-a) ©„ 
 
 correct to the first order.f 
 
 37. We have now obtained three results, Z7„ 8^, ©,, 
 as solutions to the first order of our differential equations, 
 and we must employ them to obtain the next approximate 
 solutions ; but, before they can be so employed, they will 
 require to be slightly modified — in such a manner, however, 
 as not to interfere with their degree of approximation. 
 
 The necessity for such a modification will appear from 
 the following considerations : 
 
 * When a function of a Tariable contains periodical terms which go through 
 all their changes positive and negative as the variable increases continuously, 
 the mean value of the function is the part which is independent of the periodica] 
 terms. 
 
 t We shall also employ this method of correcting the integral in our next 
 approximation to the value of 6 in terms of t ; and if we purposed to cany our 
 approximations to ■> higher order than the second, we should still adopt the 
 same value, that is, zero, for the arbitrary constant introduced by the integration. 
 To shew the advantage of thus correcting with respect to mean values : suppose 
 we reckoned the time from some definite value of 6, for instance when 8 = 0; 
 then, in the first approximation, 
 
 = t + 2e sin (t — a) 
 is the equation for determining the constant t, and in the second apjH'Oximation 
 < would be found from 
 
 = £ + 2«sin {t-a) + ie'sm 2 (i-a)+ , 
 
 giving different values of i at each successive approximation. 
 
MODIFICATION OP FIRST SOLUTION, 35 
 
 Suppose we proceed with the values already obtained; 
 we have, by Art. (24), 
 
 Fr.= r,Cl-fs)-,jTi-j-&c 
 
 = a(l-fO-^.{l+ecos(5-a)P 
 
 = a+ + A coa(6- a)+ ; 
 
 and this being substituted in the differential equation (A) 
 of Art (31;, gives 
 
 T2. + M = a+ + ^co8(^-a)+ , 
 
 the solution of which is 
 
 M = a {1 + e cos(^ - a)} + + iA0 8in(0 - a). 
 
 Our first approximate value u = a{l + e co8(5 — a)} is here 
 corrected by a term which, on account of the factor 6, admits 
 of indefinite increase, and thus becomes ultimately more 
 important than that with which we started. Such a cor- 
 rection is inadmissible, for our first value is confirmed by 
 observation (22) to be very nearly the true one. The moon's 
 distance, as determined by her parallax, is never much less 
 than 60 times the earth's radius; whereas this new value 
 of u, when 6 is very great, would make the distance in- 
 definitely small. On the same principle, we see that any 
 solution, which comprises a term of the form ^^ sin (^ — a), 
 cannot be an approximate solution except for a small range 
 of values of 6. 
 
 Snch terms, ' if they really had an existence in our system, most end in its 
 ' destruction, or at leaat in the total subversion of its present state ; but when 
 'they do occur, they have their origin, not in the nature of the differential 
 ' equations, but in the imperfection of our analysis, and in the inadequate repre- 
 'sentation of the perturbations, and are to be got rid of, or rather included in 
 ' more general expressions of a periodical nature, by a more refined investigation 
 • than that which led us to them. The nature of this difficulty will be easily 
 'miderstood from the following reasoning. Suppose that a term, such as 
 'a sin (AB + B), should exist in the value of u, in which A being extremely 
 
36 LUNAR THEORY. 
 
 ' minute, the period of the inequality denoted by it would be of great length ; 
 ' then, whatever might be the value of the coefficient o, the inequality would 
 ' still be always confined within certain limits, and after many ages would return 
 * to its former state. 
 
 'Suppose now that our peculiar mode of arriving at the value of « led us 
 ' to this term, not in its real analytical form a sin {A6 + B), but by the way of 
 ' its development in powers of 6, a + jSfl + ifl" + ic. ; and that, not at once, .but 
 ' piecemeal, as it were ; a first approximation giving us only the term a, a 
 ' second adding the term /30, and so on. If we stopped here, it is obvious tha); 
 'we should mistake the nature of this inequality, and that a really periodical 
 'function, from the effect of an imperfect approximation, would appear under 
 
 'the form of one not periodical These terms in the value of u, when they 
 
 'occur, are not superfluous; they are essential to its expression, but they lead 
 'us to erroneous conclusions as to the stability of our system and the general 
 ' laws of its perturbations, unless we keep in view that thep are only parts of 
 'strUs; the principal parts, it is true, when we confine ourselves to intpryala 
 ' of moderate length, but which cease to be so after the lapse of very long times 
 ' the rest of the series acquiring ultimately the preponderance, and compensating 
 ' the want of periodicity of its first terms.— Sir John Herschbl, Encyclopadia 
 MetropolUana — Phtsioal Abtronomy, p. 679. 
 
 38. To extritcate ourselves from this difficulty, and 
 to alter the solution so that none but periodical terms 
 may be introduced, let us again observe that the equation 
 
 ^, + M = T-s, = a, which gave the solution ZTj and thus led 
 
 to the difficulty, is only an approximate form of the first 
 order of the exact equation. Any value of u, therefore, 
 
 ■which satisfies the approximate ^equation -j^, + M = a to the 
 
 do 
 
 first order, and which evades the difficulty mentioned above, 
 
 may be taken as a solution to the same order of the exapt 
 
 equation. 
 
 These conditions will be satisfied if we assume 
 
 « = a {1 + e cos(c^ —."■)} j 
 
 then "52, +M = ffl + a€(l -c')cos(c^ — a) 
 
 = a to the first order, 
 provided 1 — c' bp of the first order at least. 
 
MODIFICATION OF FIRST SOLUTION. 37 
 
 39. The introduction of the factor c in the argument is 
 an artifice due to Clairaut ; and its effect, as we shall see 
 in Art. (66) is equivalent to supposing that the apse of the 
 moon's orbit is not fixed. Now, although the observations 
 recorded in Art. (22) did not suggest this, we must bear in 
 mind that they were extremely rough, and carried on only 
 for a short interval ; but when they are made with a little 
 more accuracy, and extended over several revolutions of the 
 moon, it is soon found that both her apse and the plane 
 of her orbit are in constant motion. 
 
 Clairaut was fully aware of these motions, and there is 
 no doubt that he was led to the above form of the value 
 of u by that consideration, and by his acquaintance with the 
 results of Newton's ninth section, which, when translated into 
 analytical language, lead at once to the same form.* 
 
 We might, therefore, taking for granted the results of 
 observation, have commenced our approximation at this step, 
 and have at once written down u = a{l + e cob{c0 — a)} ; but 
 we should, in so doing, have merely postponed the diflBculty 
 to the next step, since there, again, as we shall find, the 
 differential equation is of the form 
 
 -j^ + tt = a function of 0, 
 do 
 
 • Kewton has there shewn, that if the angular velocity of the orbit be to 
 that of the body as G - F to G, the additional centripetal force is — ^ — h'u', 
 the original force being /i«'. Therefore 
 
 d'u F^ _!i _^2J!- -El 
 am '^ G'"~h'~ G' h? ~ G" "' 
 
 F 
 
 where -^ is the same as onr c. 
 G 
 
 u = ajl +ecos(— 6 — ojj., 
 
38 LUNAR THEORY. 
 
 the correct integral of ■which would be 
 
 u = AcQa{6-B)+ , 
 
 and this would at the next operation bring in a term with 
 6 for a coeflScient, which we now know must not be. We 
 shall, therefore, hereafter omit such terms as A cos{0 — B) 
 altogether, and merely write 
 
 u = a{l + ecos(cd- a)} + 
 
 40. The value of c will be found more and more cor- 
 rectly at each successive approximation, by always assuming 
 these terms as the first two terms of the value of m, then 
 substituting in the differential equation and equating coeffi- 
 cients. It will thus be found that ae (1 - c') must equal the 
 coefficient of cos{cd — a) in the differential equation ; and this 
 will enable us to determine c to the same degree of approxi- 
 mation as that of the differential equation itself. See 
 Arts. (52) and (103). So far, all that we know about c is that 
 it differs from unity at most by a quantity of the first order. 
 
 41. In carrying on the solution of s, the same difficulty 
 arises as in u, and it will be found necessary to change it into 
 
 s = ksiD(,gd — 7), 
 g being a quantity which differs from unity at most by a 
 quantity of the first order. 
 
 The introduction of g is connected with the motion of the 
 node in the same way as that of c is with the motion of the 
 apse (85) ; and the value of g will be determined by a process 
 similar to that explained above for c. Arts. (51) and (1U2). 
 
 42. The connexion between d and t will also be modified 
 by this change in the value of m, 
 
 dt_ 1 1 
 
 de ~ Aa' {1 + e cos(ce/ - a)]' ' 
 
 d.ct J. 1 
 
 °^ dTcd ~ h^' [l + e cos{cd - a)\' ' 
 
MODIFICATION OF FIRST SOLUTION. 39 
 
 Here c0 and ct hold the places which 6 and t occupied 
 in (35) ; therefore 
 
 c0 = cpt + 2e ain(cpt — a), 
 or 6=pt + '2e sin {cpt — a), 
 
 to the first order, since - = e to the first order. 
 ' c 
 
 43. Since the disturbing forces are to be taken into 
 account in the next approximation, we shall have to use 
 the value of u found in (18), which is 
 
 M' = a'{l+e'cos(e'-f)}: 
 
 but this introduces ff ; we must therefore further modify it 
 by substituting for ff its value in terms of 5, and it will 
 be found sufiicient, for the purpose of the present work, 
 to obtain the connexion between them to the first order, 
 which may be done as follows : 
 
 Let m be the ratio of the mean motions of the sun and 
 moon, 
 
 p\ p their mean angular velocities; .'.p =Tnp, 
 
 p't + ^,pt mean longitudes at time t, /3 being the sun's 
 
 longitude when < = 0, 
 
 6\ 6 true longitudes at time t, 
 
 5", a longitude of perigees when < = ; 
 
 therefore 0' — ?= sun's true anomaly, 
 
 and p't + fi- ?= mean anomaly. 
 
 But, by Art. (13), 
 true anomaly = mean anomaly + 2c' sin (mean anomaly) + &c. ; 
 
 therefore 0'= p't-^ + 2e' sin (/ « + /3 - ?) + 
 
 = mpt-\-0 +2e' Bin [mpt+0-^)+ 
 
 = m0 + +2e'8in(»n^ + /S-?) 
 to the first order ; 
 
40 LUNAR THEORY. 
 
 because pt=6 — 2e sin (cd - a) to the first order by (42). 
 Whence u' = a'[l+e cos {md + /3 - ?)} to the first order. 
 
 44. The values of sin 2 (d- ff) and cos 2 (0-6') can also 
 be readily obtained to the same order : 
 
 sin 2(^-0') 
 
 = sin {(2 - 2m) 5 - 2/3 - 4e' sin {m9 + ^- f)} 
 
 = Sim[{2 -2m) e -20}-4:e' sia{md+^ - ^) cos{{2-2m)0-2^} 
 
 = sin{(2-2»i)0-2y8}-2e'sin{(2-ni) 0-/3- ?j 
 
 + 2e' sin {(2 - 3m) - 3/3 + ?j. 
 Similarly, cos 2 (6 - 6') = cos {(2 - 2m) - 2/3} 
 - 2e' cos {(2 - m) - /3 - ?} + 2e' cos {(2 - 3m) - 3/3 + ?}. 
 The first term of each of these is all we shall require. 
 
 Section III. 
 
 To solve the equations to the Second Order. 
 
 45. Let us recapitulate the results of the last approxi- 
 mation. 
 
 « = a {1 + e cos(c0 - a)), 
 
 u = a' {I + e' co8(m0 + j8 - ?)}, 
 
 s = isin((70— 7), 
 
 e - e'-{\ - m)e - ^ -2e' i\n{me + ^- ^). 
 
 These values must now be substituted in the expressions 
 for the forces, retaining terms above the second order, when, 
 according to the criterion of Art. (30), they promise to be- 
 come of the second order after integrating. 
 
SOLUTION TO THE SECOND OKDER. 41 
 
 The diflferential equations [A) and (5) •will then assume 
 the forms 
 
 and the integration of these will enable us to obtain u and s 
 to the second order; after which, equation (C) will give the 
 connexion between 6 and t to the same order. 
 
 46. The quantity -p-g- , which we shall meet with as a 
 
 coeiBcient of the terras due to the disturbing force, can be 
 replaced by m'^a, m being the ratio of the mean motions of 
 the sun and moon. 
 
 So long as we neglected the disturbing force, k and a 
 had determinate values : — they belonged to the ellipse which 
 formed our first imperfect solution, and would therefore be 
 known from the circumstances of motion in that ellipse at 
 any instant ; h being double the area described in a unit 
 of time, and a the reciprocal of the semi-latus rectum. It 
 would consequently be impossible to assume any arhitrary 
 connexion between them. But, when we proceed to a second 
 approximation and introduce the disturbing force, there is 
 no longer a determinate ellipse to which the h and a apply : 
 the equation /* = K'a of Art. (32) merely shews that a and h 
 must refer to some one of the instantaneous ellipses which 
 the moon could describe about the earth if the disturbance 
 were to cease, and we are at liberty to select any one of 
 these which will allow us to proceed with our approximation. 
 The particular ellipse is determined by the above assumed 
 
 relation -rr^ =m'a, and the selection is suggested and 
 A a 
 
42 LUNAR THEORT. 
 
 justified by the following reasoning : 
 
 jKi' average period of moon about earth 
 p period of sun about earth ' 
 
 but, since the instantaneous ellipses are nearly circles, we 
 have, as in Art. (25), 
 
 (period of moon about earth)* _ wi'a" . 
 (period of sun about earth)' ^a' ' 
 
 therefore if a be properly chosen, 
 
 f rs f rs 
 
 , ma tna 
 m = — ,- = ,2 4 • 
 /*a A a 
 
 47. If we examine the equations of Art. (31) it will be 
 seen that we cannot solve the differential equation in u to 
 the second order until we know the value of a accurately to 
 that order ; for, suppose 
 
 s = A;sin(5r^-7)+ F„ 
 
 where F^ contains the terms of the second order not yet 
 found; and suppose that among the terms in F, there be 
 one L sin (Z^ — X), then this value of s substituted in the 
 expression (Z)) gives 
 
 ^ = ;^.{l-f[^-8i°(^^-7) + isin(?^-\)...]V- 
 
 = a {1 - \ie am'igd-r^) - 3kL sin (^5 - 7) s\q[W-\)...], 
 and equation (A) becomes, so far as tliis term is concerned, 
 
 KS + ") = ^ +••- ^^^ t'°' ^^^ -i)e-{y- \)]] 
 
 + ?,kL[coa[(g + T)e-[y + \)}].... 
 
 Now, if It should happen that ? is a small quantity, then 
 the two coefficients g — I and g-\- I will be both near unity. 
 
SOLUTION TO THE SECOND OHDER. 43 
 
 and therefore these two terms, although of the third order 
 in the differential equation, would become important in the 
 value of M. Again, if I is near 2, then l-g will be near 
 unity, and the same remark will apply. 
 
 Such terms do not really occur in the value of s, but 
 there is none the less a necessity for verifying the fact. 
 
 If I is near unity, then the coefficient g-l becomes small ; 
 and although the corresponding term in u is still of the third 
 order, it becomes of the second order, and therefore impor- 
 tant in t, Art. (30). 
 
 If I is not near 0, 1 or 2, the term will not aflfect, to 
 that order, the values of u or of t. 
 
 48. Again, it is necessary to have the value of u correct 
 
 T 
 
 to the second order, before we can compute 7^-3 to the fourth 
 
 order, with a full assurance that we have secured all terms 
 of that order in which the coefficient of the argument 
 is nearly zero* ; — such terms becoming, as we have seen 
 Art. (30), important in the value of t. 
 For, assume that 
 
 u = a{l+e coi[c9 - a) + PTJ, 
 
 = - f m' (1...- 4 PFJ sin {(2 -2m) 6- 2(S}. 
 
 Therefore if W^ contain a term B cos{rd — p) of the second 
 order where r is near 2 (except 2 — 2m) there will be a 
 
 * This was pointed oat by Mr. 'Walton in the Quarterly Journal of Mathematict, 
 Vol. II., p. 227 ; and in this edition the text had been modified in accordance 
 with his suggestion ; but the method given by him for finding the value of u is 
 defective on account of its not mentioning the necessity for the previous deter- 
 mination of «. 
 
44 LUNAR THEOKY. 
 
 T 
 corresponding term in p— 3 of the fourth order with a small 
 
 coefficient of 6, and therefore an important term in the 
 value of t. 
 
 49. For the sake of simplifying the expressions we shall 
 generally omit a, yS, and 7 in the arguments. If we remark 
 that c and a always appear together in the form c0 — a, it 
 will be sufficient to write c^, instead of c^ — a ; and, for 
 the same reason, we may replace md + ^ and gO — y by 
 mdj and gO^. The suffix (J will remind us of the omission, 
 and we may at any stage re-introduce the omitted symbols. 
 
 To commute s to the second order. 
 
 50. Since we must first determine the value of s, it 
 becomes necessary to examine the differential equation {B) 
 to see whether any difficulty of a similar kind to those we 
 have been here considering may arise, or whether the 
 elements on which the computation depends are already 
 known with sufficient accuracy. We fortunately find that 
 all the terms of the right-hand member of the equation are 
 of the third order, and therefore that the values already 
 obtained will ensure accuracy to the second order, and we 
 may safely begin our work here. 
 
 51. To the first order we have 
 
 s = k aingd^j 
 
 ds J /I 7 /, (because a=l 
 
 ^=''^'=°^^^« = *^°^^^'lto first order, 
 
TO COMPUTE S TO THE SECOND ORDER. 4.0 
 
 Ps— 8 
 In the expression for ^ ■ , ■which is of the third order, 
 
 it will be sufficient to put a for u, and a for m, then 
 Ps-S , OT'a" , . „ , 
 
 = — Im'i- sln_^0, + ^rri'k sin (2 - 2»i — ^) ff^, 
 
 the other term of the third order being rejected because the 
 
 coefficient of 6 is not near unity, 
 
 T ds ds 
 
 The next term of equation {B) is Ta-i^a) ^"d as ^^^ is 
 
 T 
 of the first order, it will be sufficient to have -^r^^ correct 
 
 to the second order, 
 
 T 
 ^3 = -|™'sm(2-2m)^„ 
 
 T ds 
 ■p-j •-fa — — ^i^k sin (2 — 2m) 6^ cosgd^ 
 
 = - Im'k sin (2 -2m-g) 0^, 
 
 neglecting the other term of the third order because the 
 coefficient of 6 is not near unity. 
 
 f T 
 which =0 to the third order, since \j—^dd depending on 
 
 the disturbing force is of the second order, and the other 
 factor = to the first order. 
 
 The equation in s becomes 
 d's 
 
 d¥'^' = 
 — ^m'k singd^ 
 + |rn'i sin (2 -2m- g) 6^. 
 
 To solve it, assume 
 
 s = 
 
 kamgO^ 
 + -4 sin (2 -2m -5') 5,. 
 
46 LUNAR THEORY. 
 
 Substitute, and equate coefiScients of like terms, 
 k{l-g'] = - ^m'k ■ .:g = {l + ^m')i = 1 + K 
 
 Therefore, accurately, to the second order,* 
 
 s = k am{g0 - y) + !^mk sin{{2 -2m- g) -2^ + 'y}...S,. 
 
 The term of the second order, having a coefficient near 
 unity, need not be taken into account in finding the value 
 of u (47) ; but it will have to be brought in subsequently 
 to determine t. 
 
 To determine u to the second order, 
 
 52. We must compute the right-hand member of the 
 equation (A), Art. (31), accurately to the second order, and 
 include those terms of the third order which have a coefficient 
 near unity. 
 
 To the first order we have 
 
 u = a{l + e coac^,), 
 
 du . - . „ (because c = 1 
 
 -77, = — aecsmcp, = — aesmcp, i /. . , 
 dti ' ' (to first order, 
 
 d'u 
 
 From the expressions (2>) and (£), Art. (31), we get 
 
 -fi'sin'sr^.) 
 
 Wa"{l+e'cos(.g-r))' 3 _ 
 
 2AV (1 + e co8c6>,)" ' ^ ' '" 
 
 ^, = a(l-|A:'8in'sr^.) 
 
 • No complementary term P cos (6 — Q) is added ; for, though by the theory 
 of differential equations, this would form a necessary part of the solution, we 
 have seen Art. (39) that it cannot, in this shape, form a part of the conect value, 
 but will be comprised In the terms whose argument is g9 — y. 
 
TO DETERMINE M TO THE SECOND ORDER. 
 
 47 
 
 - ^wi" {1 + 3e' cos{me^ - ?) - 3e cosc^,} (1+3 cos (2 - 2m) 0j, 
 
 1 F 
 
 - ^ = 1 - 1 A' + f /c' cos 2g0^ - f m' - fm' cos (2 - 2m) ^, 
 
 + f m'e cosc^j + |m'e cos (2 — 2?w — c) 0,. 
 The other terms of the third order, which arise in the 
 development of the expression, are neglected because the 
 coefficients of are not near unity. 
 
 T _ 3m'a"{l + e'cos{me^-^)Y 
 
 8in(2-2»i)^, 
 
 hW 2rV(H-ecosc^J* 
 
 = - fw' [1 + 3e' cos [md, - ?) - 4e cosc^,] sin (2 - 2m) 6^ 
 = - |m' sin (2 - 2m) 6^ + Zm'e sin (2 - 2ot - c) ^„ 
 
 T.',.i • M) = i"*'"* ^'° (2 ~ 2'") ^1 si°c^, 
 
 = |m'ac cos (2 — 2»i — c) ^, ; 
 rejecting the other terms of the third order according to (30). 
 
 /; 
 
 cos(2-2jn)^,-, 
 
 Swi'e 
 
 cos(2-2m-c)^, 
 
 AV *2-2wz ''" —'-^ 2-2m-c 
 
 = Im' cos(2 - 2m) 0, - Sm'e cos(2 - 2»n - c) ^„ 
 
 2 (^+m) r,-r-,<?^=i»i'acos(2-2m)0-6«i'aecos(2-2»n-c)^,. 
 
 The equation in m becomes 
 1 /cPu \ 
 
 -a[de' + V = 
 
 + fjn"''ecosc0, 
 
 + fA;'cos29^, 
 
 -3jn'cos(2-2m)0, 
 
 + »/»n'e cos (2 - 2ot - c) 5,. 
 
 To integrate this, assume 
 
 M _ 
 
 a 
 
 1+A 
 + ecosc0^ 
 + B cos2gd^ 
 + Cco8(2-2ffi)5, 
 + Z)co8(2-27n-c)5,. 
 
48 
 
 LUNAR THEORT. 
 
 Substitute, and equate the coefficients of like terms : 
 e (1 — c') = fffi'e, 
 
 C {1 - (2 - 2m)'} = - 3m% 
 
 c=(l-fm')i=l-fOT', 
 
 — 4A. , 
 
 * 1-4 
 
 n 3m' , , 
 
 D= 
 
 1-4 
 15m'e 
 
 2[l-(l-2»i+f»i'j 
 
 Z){1-(2-2ot-c)'}= V«'e. 
 Therefore 
 
 + e cos (c^ — a) 
 u = a i -;ii'co32 (gff-y) 
 
 + m'cos{(2-2m)d-2l3] 
 ^ + Vme cos {(2 -2m- c) d-2^ + a} 
 This is the accurate value of u to the second order. 
 
 To find t to the second order. 
 
 53. The equation to be considered is 
 dt 1 / ^rT 
 
 TTfi = ¥»»«• 
 
 .U. 
 
 dt I (, ^[T ja\~^ 
 
 and, making use of the values of s and u obtained to the 
 
 second order, we must retrace our steps to recompute u and 
 
 T 
 yj-, so as to include those terms of the third order in w in 
 h u 
 
 v?hlch the coefficient of the argument is near zero, and 
 
 T 
 similar terms to the fourth order in ^.-^ Art. (48). To do 
 
 h u ^ ' 
 
 this we need not repeat the whole work, but only so much 
 of it as will enable us to pick out the additional terms in 
 question. 
 
TO FIND t TO THE SECOND ORDER. 49 
 
 a AV " ^ " * ^* sin^^, + |mA sin {2 -2m- g) ^J' 
 
 -i»n'{l+3e' co8(w0 -?)-3e cosc0,}{l+3 cos(2-2»i)^/( 
 = l...-|7wA;' 6\agd^ sin (2- 2m- g) 6- f thV cos(»n0 -?) 
 = 1...- -^mie cos(2 - 2wi - 2^) 0^ - f wV cos(«»^, - ?), 
 r _ 3m'a" (1 + e coswg, - Q' sin (2 - 2m) g, 
 AV ~ 2A'a* (1 + e cosc^, - J/fc' co325'd^,)* 
 
 = - f m' sin (2 - 2»n) 0,{l+e coacd^ - \ie cos2^0J "* 
 = - f m" sin (2 - 2m) ^, {(1 + e cosc0,)~* + /:' CQ%2g6^ 
 = -fni' 8in(2 -2»?) 0, {...lOe' cos'c5, + /fc' cos2^^,} 
 = - f »i' sin (2 - 2m) 6^ {...be* eos2c0, + A* eo5f2j9r0J 
 = _ i^,MVsin(2-2»»-2c)0,- frnT 8in(2-2m-2^) ^,. 
 These are the onlj important additional terms. 
 
 • T 
 
 ,i-id6=- ^m^ cos (2-2WI -2c) 0-^mJt^ C03(2-2m-2g) ^„ 
 
 because 2-2m-2c=-2m + ^m', and 2-2m-2g=-2m—^m^. 
 Substituting in the diflferential equation for u, the new 
 
 rp J 
 
 terms in tt-i t^ '"ill be of the fourth order arrd must be 
 AV dd 
 
 -j2j + m) |7r-i<^^ ■will bring in the two 
 
 terms 
 
 — »^»wc'acos(2-27«-2c) ^, and -fmA;'ocos(2-2OT-25r)^,; 
 and we get, so far as important additional terms are concerned, 
 
 \ J|^ + m1 = 1...+ V»ne'cos(2-2OT-2c)^, 
 
 + (_ ^ + 1) mF cos (2 - 2»i - 2g) 6- f mV cos (mO- ?), 
 and the corresponding terms in u will also be of the third 
 order, viz. 
 
 - =. . . Y me" cos (2 - 2m - 2c) 5, + r^mk" cos (2 - 2m - 2g) 6^ 
 
 CL 
 
 - fmV cos(m^,-?). 
 
 /, 
 
50 LUNAR THEORY. 
 
 The complete value of u therefore, so far as important terms 
 are concerned for determining the value of t, is 
 
 ? = 1 _ |yt' - ^«i' + e cosc^i - P' cos 2^0, 
 d 
 
 + to' cos (2 - 2m) 0^ + V»n« cos (2 - 2»n - c) 0^ 
 
 + 1/to' cos (2 - 2to - 2c) 0, + ^mk" cos (2 - 2m - 2^-) 0^ 
 
 — ^rrt'e cos(m0^ — f), 
 
 f T 
 and the complete value of Jtj-, d0 will be 
 
 Itt-s <?^ = f «i' cos (2 - 2m) 0, - V"* cos (2 - 2»n - 2c) ^, 
 - fmi' cos (2 - 2m - 2^) ^„ 
 
 the other term of the third order has no influence on <, the 
 coefficient of not being small. 
 
 54. We are now in a position to find the value of t to 
 the second order : 
 
 d0 
 
 -M'^'&''y=M'-i^''') 
 
 If - = 1 + ^ COSC0, + W where W contains all terms of the 
 a 
 
 second and third orders in the expression above, then 
 J-, = l,(l+ccosc^. + Tr)- 
 
 = ^, {1 - 2 (fi cosc^i+TF) + 3 (e cose^^+TT)'-...} 
 
 = ^, (1 - 2e cosc^j - 2 W+ Se* cos'c^, + 6e TTcosc^,)... 
 
 = ^. (1 + |e' - 2tf COSC0 - 2 W+ |e' cos2c0,+ 6e TFcosc^,). 
 
TO FIND t TO THE SECOND ORDER. 51 
 
 The only part of GePTcosc^, which gives a term of the third 
 order in which the coefficient of the argument is small, is 
 
 6ecosc^, {...+ i^me cos(2- 2wi-c)^, ...}, 
 of which one term is 
 
 + V»»e' cos (2 - 2»« - 2c) d^. 
 Therefore -j—t is equal to 
 
 (1 + fe' + %Ji' + m*-'2e coscd^ + f e' cos2c^, 
 + \k' cos 2^0, - 2m' cos (2 - 2ot) ^, 
 - "^me cos(2 _ 2»i - c) 0, + 3»jV cos(m^, - f) 
 - »/me' cos (2 - 2?m - 2c) 6>,- %mk' C08(2 - 2ot - 2^) ^,. 
 Also from above 
 
 J _ /■_y^^^^fl-f»i'cos (2-2»i) e^+^me' cos (2-2ot-2c) d^ 
 J h'u' \ + ^mk' cos (2 - 2ot - 2g) ^,. 
 
 We have now to multiply these results together, and we see 
 
 that the terms having for coefficients ^me' and fmi" will 
 
 disappear in the product.* These were the terms which 
 
 . . T 
 
 were originally of the fourth order in rr-i • 
 
 (1 + f e' + f A^ + Ml' - 2e cosc^, + f e' cos2c5, 
 -I- ^k' cos2^5, - V^" cos(2 - 2m) 6^ 
 — ^me cos (2 -2m-c)^^■>^ Sjw'e' cos(»n5, - f). 
 
 ♦ The principle of the superposition of small disturbances might have led us 
 a priori to anticipate the disappearance of these terms: — for, when approximating 
 to the second order we may, according to this principle, calculate separately the 
 effect of the disturbing action of the sun, which is of the second order, and that 
 of any other independent disturbance of the same or of a higher order, and then 
 add the results. No term therefore which bears a trace of the action of both 
 these forces should present itself to the second order. 
 
 Now since neither k nor e enters into the argument of any term, it is obvious 
 that a term whose coefficient contains /c'' or e' at any stage will retain them 
 through all subsequent operations ; and if the terms having (2 — 2m — 2g) 9, 
 or (2 - 2m — 2c) 8, for argument could remain in the result to the second order 
 
52 LUNAR THEORY. 
 
 Let l.,(i + |e'' + |A' + «')=^, 
 
 therefore f = Aa' (I — \i — p' - »n') to the third order ; 
 therefore, multiplying by p and integrating, we get, still to 
 the second order, 
 
 ^<=0-2esin(c5-a) + |e*8in2(c5-a) 
 + jA'sin2(.90-7) 
 
 - V»n'8in{(2-2ni)5-2^} )...©„ 
 
 - Y'w«sin((2-27n-c)^-2y8+a} 
 + Swze' sin (to^ + )S - f ) 
 
 no constant is added, the time being reckoned from the 
 instant when the mean value of Q vanishes, for the reasons 
 explained in (Art. 36). 
 
 Coordinates expressed in terms of the time. 
 
 55. The preceding equations U^, jS,, 0, give the reci- 
 procal of the radius vector, the latitude and the time in 
 terms of the true longitude ; but the principal object of the 
 analytical investigations of the Lunar Theory being the 
 formation of tables which give the coordinates of the moon 
 at stated times, we must express u, s, and 6 in terms of t. 
 
 To do this, we must reverse the series pt=d — &c., and 
 then substitute the value of 6 in the expressions for u and s. 
 
 Now d—pt + le sin (c6 — a) to the first order 
 
 = pt+2e ^\n{cpt — a) ; 
 
 they would there have i' or e^ respectivdy in the coefficient, since they have 
 them when they first appeal". Bnt A^, «^ and m being independent, such terms 
 must have arisen from the combiaation of others which originally (in the 
 expressions of the forces) involved these quantities separately: — terms like 
 i^ cos 2 (gd — y) and e' oos 2 {c6 — a) combined with terms involving m : that 
 is, terms which are representatives of disturbing forces of the second order (see 
 note p. 31) combined with others depending on the sun's action. Such terms 
 must therefore be of an order beyond the second. 
 
COORDINATES EXPKEBSED IN TERMS OP THE TIME. 53 
 
 therefore cd — a = cpt — a + '2e sin (c.pt — a) to the first order, 
 
 2e sin (cd — a) = 2e (sin (ept — a) + 2e sin (cpt — a) cos (cpt — a)} 
 
 to the second orderj 
 
 = iesin(cpt — a) + 2e'sia2 (cpt— a) ; 
 
 and as and pt differ by a quantity of the first order, they 
 may be used indiscriminately in terms of the second order ; 
 therefere 
 
 ff=^t+2e sin {cpt — a)+ f e* sin 2 (cpt — a) 
 
 — JA' sin 2 {gpt — 7) 
 
 + V'n'sin{(2-2»n)pt-2y3} ...0',, 
 
 + ^me sin {(2 - 2m -c) pt - 2/3+a} 
 
 — Swie' sin (mpt + /S — §■) 
 
 56. In the value of u given in Art. (52), substitute pt for 
 6 in terms of the second order, and pt + 2e sia{cpt - a) in the 
 term of the first order ; then 
 
 l—^k'-^m'—e'+ecos(cpt-a)+e'cQs2(cpt-a.) 
 
 — ^k* cos2 (gpt - 7) 
 + m' cos ((2 - 2m) pt - 2/3} 
 + ywie cos {(2-2m-c)pt-2^ + a] 
 
 u = a[ 
 
 ..u:. 
 
 57. Similarly, the expression for s becomes 
 s = k sin [(jjpt - 7) + 2e sin {cpt - a)} 
 
 + %mk sin {(2 -2m- g)pt - 2^8 + 7} ; 
 
 im(gpt-'i) 
 ^^^\^e&{Q[{g + c)pt-a-'i] ^,_ 
 
 or 
 
 - e sin \{g - c)pt + 0-7} 
 
 + fw sin {(2 -2m-g)pt-2^+ 7} 
 
 The expression for s is more complex in this form than when 
 
 given in terms of the true longitude 6. 
 
54 LUNAR THEORY, 
 
 Moon's Parallcix. 
 
 58. If P be the moon's mean parallax, and n the parallax 
 at the time t, 
 
 _ radius of earth R _ ., i in^ .. ^i- j j 
 
 "= distance of D = 17—=^^ -^.O to the third order, 
 
 = iZw {1 — JA' + ^f A;' cos2 {gpt — 7)} to the second order, 
 
 11 - A' — I w' — e' + e cos (cp< — o) + e' cos2 (cpt — a) 
 + m' cos j(2 - 2m);}< - 2/3} 
 + 1^5 wie cos {(2 - 2?H - c)^< - 2/3 + a} ; 
 but P= the portion which is independent of periodical terms, 
 
 11 + e cos (cpt — a) + e' cos 2 (cpt — a) 
 + to" cos 1(2 - 2m) pt - 2/8} 
 + *^7necos{(2-2OT-c)pf-2/3 + a} 
 
 neglecting terms of the third order. We see that, to the 
 second order, the variable part of the parallax is independent 
 of the inclination. 
 
 59. Here we terminate our approximations to the values 
 of M, s, and 6. If we wished to carry them to the third 
 order, it would be necessary to include some terms of the 
 fourth and fifth orders according to Art .(30), and the ap- 
 proximate values of P, T, and S^ given in Art. (24), would 
 no longer be suflSciently accurate, but we should have to 
 recur to the exact values, and from them obtain terms of 
 an order beyond those already employed.* The process 
 followed in the preceding pages is a sufficient clue to what 
 would have to be done for a higher approximation. 
 
 * See Parallactic Inequality, Art. (105). 
 
SOLUTIONS TO A HIGHER ORDER. 55 
 
 The coordinates u and & of the sun's position are, by 
 the theory of elliptic motion, known in terms of the time i, 
 and t is given in terms of the longitude B by the equation 
 0j. Hence m' and & can be obtained in terms of B\ but 
 it will be necessary to take into account the slow progressive 
 motion of the sun's perigee, which we have hitherto neglected. 
 This we may do by writing cB' — f for B' — f, e' being a 
 quantity which differs very little from unity.* 
 
 These values of m', B\ together with those of w and s 
 given by U^ and iS„ are then to be substituted in the 
 corrected values of the forces, and thence in the differential 
 equations. The integrations being performed as before will 
 give the values of m, s, and t in terms of 6 to the third 
 order, and from these, as in Arts. (55), (56), and (57), may be 
 obtained u, s, and B in terms of t. 
 
 60. More approximate values of c and g are obtained at 
 the same time, by means of the coeflScients of cos {cB — a) 
 and sin {^B — 7) in the differential equations, (see Appendix, 
 Arts. 102 and 103). 
 
 61. The values to the fourth order are then obtained 
 from those to the third by continuing the same process, and 
 
 • 'En reflechisBant sor les termes que doivent introdnire toutes les quantitea 
 • pi&:edentee, on voit qu'il se pent glisser des cosinos de I'angle 6 dont nous avons 
 < Tn le dangereux effet d'amener dans la Taleur de u des arcs au lieu de leuis 
 'cosinus; de tels tennes viendront, par exemple, de la combinaison des cosinus 
 'de (1— in) 6 avec des cosinus de mfl 
 
 < Pour eviter cet inconvenient qui Sterait ^ la solution precedente 
 
 'I'aTantage de convenir i, un aussi grand nombre de revolutions qu'on voudrait, 
 ' et la priveiait de la simplicite et de 1' universalite si precieuses en mathgmatiques, 
 'il faut commencer par en cheroher la cause. Or, on deoonvre facilement que 
 'ces tennes ne viennent que de ce qu'on a suppose fixe I'apogge du soleil, ce 
 ' qui n'est pas permis en toute rigueur, puisque quelque petite que soit sur cet astre 
 ' Taction de la lune, eUe n'eu est pas moins reeUe et doit lui produiie un mouvement 
 ' d'apogSe quoique trgs lent i. la vfirit^'. — Clairaut, Tlieorii de la Lune, p. 55. 
 2me Edition. 
 
56 LUNAR THEORY. 
 
 80 on to the fifth and higher orders ; but the calculations are 
 so complex that the approximations have not been carried 
 beyond the fifth order, and already the value of 6 in terms 
 of t contains 128 periodical terms, -without including those 
 due to the disturbances produced by the planets. The coefli- 
 
 cients of these periodical terms are functions of m, e, e\ — , 
 
 c, g, k, and are themselves very complicated under their 
 literal forms : that of the term whose argument is twice the 
 difference of the longitude of the sun and moon, for instance, 
 is itself composed of 46 terms, combinations of the preceding 
 constants. 
 
 See Pont^coulant, Systhne du Monde^ torn. IV., p. 572. 
 
( 57 ) 
 
 CHAPTER V. 
 
 NUMEBICAL VALUES OF THE COEFFICIENTS. 
 
 62. Having thus, from theory, obtained the form of the 
 developments of the coordinates of the moon's position at 
 any time, the next necessary step is the determination of the 
 numerical values of the coefficients of the several terms. 
 
 We here give three different methods which may be 
 employed for that purpose, and these may, moreover, be 
 combined according to circumstances. 
 
 63. First Method. The values of the constants p, m, a, 
 /8, 7, f which enter into the arguments, and of the additional 
 ones a, h, e, &c. which enter into the coefficients, of the 
 terms in the previous developments, may be obtained with 
 great accuracy from observation. 
 
 For this we must employ observations separated by very 
 long intervals, such, for instance, as ancient and modern 
 eclipses; also particular observations of the sun and moon 
 made when the bodies occupy certain selected positions; 
 and other observations of a special character. 
 
 When the values of the elements have been so obtained, 
 then the theoretical values of the coefficients may be com- 
 puted by substitution in the analytical expressions. 
 
 64. Second Method. Let the constants which enter into 
 the arguments be determined as in the first method ; and let 
 a large number of observations be made, from each of which 
 a value of the true longitude, latitude, or parallax is obtained, 
 
 I 
 
58 LUNAK THEORY. 
 
 together with the corresponding value of t reckoned from 
 the fixed epoch when the mean longitude is zero. Let these 
 corresponding values be substituted in the equations, each 
 observation thus giving rise to a relation between the unknown 
 constant coefficients. 
 
 A very great number of equations having been thus 
 obtained, let them, by the method of least squares or some 
 analogous process, be reduced to as many as there are 
 coeflScients to be determined. The solution of these simple 
 equations will give the required values. 
 
 This method, however, would scarcely be practicable in 
 a high order of approximation. For instance, in the fifth 
 order, as stated in Art. (61), each observation would give 
 rise to an equation containing 129 unknown quantities and 
 the immense number of equations so obtained would have 
 to be reduced to 129 equations of 130 terms each. 
 
 65. Third Method. When the constants which enter into 
 the arguments have been determined by the first method, we 
 may obtain any one of the coeflScients independently of all 
 the others by the following process, provided the number of 
 observations be very great. 
 
 Let the form of the function be 
 
 F=-4 + 5sin^+ Csin^ + &c, 
 and let it be required to determine the constants A, B, C, &c. 
 separately ; 6, <f), &c., being known functions of the time. 
 
 Let the results of a great number of observations corre- 
 sponding to values 6^, 6^, 0^, &c., <^„ ^„ ^^, &c., be F„ F„ V„ 
 &c. ; so that 
 
 V^ = J +Bsm6^+ (7sin<^j4&c., 
 V^ = A+Bsm0^+ (7sin(^^ + &c., 
 F3 = ^ + .B sin ^3 + C sin<^3 + &c., 
 
 F„ = ^ + i? sin ^,, + C sin ^„ + &c. 
 
NUMERICAL VALUES. 
 
 59 
 
 therefore 
 
 Now, n being very great, we may assume that the sum of 
 the positive values of each periodical term will be about 
 counterbalanced by the sum of its negative values ; and 
 therefore, that if we add all the equations together these 
 terms will disappear ; 
 
 n ' 
 
 which determines the non-periodic part of the function. 
 
 To determine B. Let the observations be divided into 
 two sets separating the positive and negative values of sin ; 
 then the other periodical terms, not having the same period, 
 may be considered as cancelling themselves in adding up the 
 terms of each set. Let there be r terms in the first set 
 
 and s terms in the second, and let F', F", V be the 
 
 values of F corresponding to positive values of sin 0, which 
 values we may assume to be uniformly distributed from 
 
 sin^ to sinTT, and therefore to be sin 50, 8in280, sinr.S^, 
 
 where r . 60 = tt. 
 
 And, again, let F,, F^^, F^^,, V,, be the values of F 
 
 corresponding to the negative values of sin0 ; viz., sin (- A0), 
 sin(-2A0), sin(-.9.A0), where s.A0 = 7r. Then, 
 
 V =A + B6mSd + Csin<^' +..., 
 V" = A + BsLn2.S0-{- Csin^"+..., 
 
 V' =A + Bamr.S0 + Camfji' +...; 
 therefore 
 
 V'+ V"+...+ V'=r.A+B^J'(sm0) 
 
 B0' 
 
 therefore 
 
 V'+V'+.-. + V 
 
 
 siaO.dd 
 
 = A + 
 
 25 
 
 V^=A-B&mA0 +Csin(^, 4..., 
 V^^ = A-Bsin2.Ad+ (7sin^„ + ..., 
 
 F, =^-5sins.A0+ (7sin<^, +...; 
 
 therefore 
 
 F, + F,,+...+ V.=s.A-BS„^(ame) 
 
 = sA-^g-2:{sm0.A0); 
 
 therefore 
 
 s ■n-J„ 
 
 d0 
 
 TT 
 
60 LUNAR THEORY. 
 
 ^ fV-^ ¥"+...+ ¥' F + F,+...+ F,> 
 
 therefore B ■ 
 
 4 I r 5 r 
 
 and in a similar manner may each of the coefficients be 
 independently determined* 
 
 66. When the periods of two of the terms differ but 
 slightly — for instance if and <j) go through their periodic 
 variations very nearly in the same time, — the method could 
 not then with safety be applied ; for, since the same values 
 of 6 and <}> would very nearly recur together during a longer 
 time than that through which the observations would ex- 
 tend, the two terms would be so blended in the value of 
 V that they would enter nearly as one term — the difference 
 between 6 and ^ would be very nearly the same at the 
 end as at the beginning of the series of observations. 
 
 • If ?• and s are not sufficiently great to allow us to substitute /o'^ sin 6 dS for 
 
 S,'^ sinfl . id, we must proceed as follows : 
 
 V + V" + + V=tA + B (since + sm2Se + + sinrid) 
 
 . , -sinJC + l) *9sinJr.«e 
 
 = rA + H . , ,„ — 
 
 sin iot> 
 
 . „ cos we 
 
 = rA + B ~-ii, , 
 siniJ6 ' 
 
 V'+ V + + V'- , B 1 -iie^ 
 
 = A -^ — -7-— — , .„^ , nearly, 
 
 VI l\ 
 
 {-fi(^y}(- 
 
 r'+ V" + + v y,+ r„+ + r.\ 
 
NUMERICAL VALUES. 61 
 
 67. Let us suppose the periods to be actually identical, 
 so that <f) = 6 + a, a. being some constant angle ; then 
 
 ^ sin ^ + C sin <^ 
 
 may be written {B+C cos a) sin 6+ Csina cos 0, 
 
 or V= A + [B + C cosa) sin^+ Csina cos 5+ 
 
 If now we divide the observations, as before, into two 
 sets, corresponding to the positive and negative values of 
 sin^, the terms involving cos^ will disappear in the sum- 
 mation of each set ; and, following the process of the method, 
 we shall find the value of 
 
 B-i- G cos a. = M suppose. 
 
 Dividing again into two sets corresponding to the positive 
 and negative values of cos^, the terms in sin^ will be can- 
 celled, and the same process will give the value of 
 
 C sin a = iV^ suppose. 
 
 Treating the observations in the same way with respect 
 to the angle ^, we get two results, 
 
 C+BcQia. = M', 
 
 -Bsma = N'; 
 
 from these four equations we easily get 
 
 ^_ M'N+MN' __ M' N+MN ' 
 
 AP-M" °'"~ N'-N" ' 
 
 C- ^B- 
 
 M^ N, M', N' are connected by the equation of condition, 
 
 M'-M" = N"-N'. 
 
 68. When the periods of 6 and <j) arc nearly, but not 
 exactly, the same, this equation of condition will not hold, 
 and the preceding values of B and C would not be exactly 
 
G2 LUNAK THEORY. 
 
 correct ; but yet they would be very approximate, especially 
 if the mean between the two values of B be taken. 
 
 Or, we may, after having taken one of these slightly 
 erroneous values for B, make a further correction by estab- 
 lishing as it were a counterbalancing error in the value of 
 C. Let B' be the value so found for B; then, from the 
 V of each of the observations subtract the value B' sin dy 
 the result U will be very nearly equal to .4 + C7 sin ^ + &c., 
 and from the n equations 
 
 U^ = A + (78in^,+ 
 
 U^ = A+C6m<l>^+ 
 
 Z7„ = ^+ Csin^„+ 
 
 a value C of C will be obtained, by the rule, which will be 
 very approximate, and, at the same time, agree better with 
 B' in satisfying the equations than C itself would do. 
 
 69. When two terms whose periods are nearly equal do 
 occur, it is plain, by examining the values of M and M\ that 
 the errors which would be committed by following the rule, 
 without taking account of this peculiarity, would be the taking 
 B-\- (7 cos a andC+ £ cos a for B and C respectively. 
 
( 63 ) 
 
 CHAPTER VI. 
 
 PHYSICAL INTERPRETATION. 
 
 70. The solution of the problem which is the object of 
 the Lunar Theory may now be considered as effected ; that 
 is, we have obtained equations which enable us to assign 
 the moon's position in the heavens at any given time to 
 the second order of approximation ; we have explained how 
 the numerical values of the coeflScients in these equations 
 may be determined from observation; and we have, more- 
 over, shewn how to proceed in order to obtain a higher 
 approximation.* 
 
 It will, however, be interesting to discuss the results we 
 have arrived at : — to see whether they will enable us to form 
 some idea of the nature of the moon's complex motion ; also 
 how far they will explain those inequalities or departures 
 from uniform circular motion which ancient astronomers had 
 observed ; but which, until the time of Newton, were so 
 many unconnected phenomena; or, at least, had only such 
 arbitrary connexions as the astronomers chose to assign, by 
 grafting one eccentric or epicycle on another as each newly 
 discovered inequality seemed to render it necessary. 
 
 It is true that our expressions, composed of periodic 
 terms, are nothing more than translations into analytical 
 
 * The means of taking into account the ellipsoidal figure of the earth and the 
 disturbances produced by the planets, are too complex to form part of an intro- 
 ductory treatise. For information on these points reference may be made to 
 Airy's t'iijure of the EaHh. Poutccoulaut's Sjstciiie du blonde, vol. iv. 
 
G4 LUNAU TIIEOKY. 
 
 language of the epicycles of the ancient ;* but they are 
 evolved directly from the fundamental laws of force and 
 motion, and as many new terms as we please may be ob- 
 tained by carrying on the same process ; whereas the epicycles 
 of Hipparcbus and his followers were the result of numerous 
 and laborious observations and comparisons of observations ; 
 each epicycle being introduced to correct its predecessor 
 when this one was found inadequate to give the position 
 of the body at all times : just as with us, the terms of the 
 second order correct the rough results given by those of 
 the first; the terms of the third order correct those of the 
 second, and so on. But it is impossible to conceive that 
 observation alone could have detected all those minute 
 irregularities which theory makes known to us in the terms 
 of the third and higher orders, even supposing our instruments 
 far more perfect than they are ; and it will always be a 
 subject of admiration and surprise, that Tycho, Kepler, and 
 their predecessors should have been able to feel their way 
 so far among the Lunar inequalities, v^ith the means of 
 observation they possessed. 
 
 LONGITUDE OF THE MOON. 
 
 71. We shall first discuss the expression for the moon's 
 longitude, as found Art. (55), 
 
 6 =pt + 2esin {cpt -a) + |e' sin 2 {cpt - a) 
 + '^»ne sin {(2 -2m- c)pt-20 + a] 
 + Vm' sin {(2 - 2m) pt - 2/3} 
 
 - 3/ne' sin {mpt + /3 - f) 
 
 — ^k' sin 2 [gpt— 7). 
 
 The mean value of 9 h pt; and in order to judge of the 
 effect of any of the small terms, we may consider them 
 
 ♦ See Whewcll's Ilistonj nf the Iiuhictiee Sciences. 
 
ELLIPTIC INEQUALITY. 65 
 
 one at a time as a correction on this mean value pt, or 
 we may select a combination of two or more to form this 
 correction. 
 
 We shall have instances of combination in the elliptic 
 inequality and the evection, Arts. (73) and (77) ; but in the 
 remaining inequalities each term of the expression will form 
 a correction to be considered by itself. 
 
 72. Neglecting all the periodical terms, we have 
 
 de 
 
 which indicates uniform angular velocity; and as, to the 
 
 same order, the value of u is constant, the two together 
 
 indicate that the moon moves uniformly in a circle, the 
 
 27r 
 period of a revolution being — , which is, therefore, the 
 
 expression for a mean sidereal month, or about 27J days.* 
 The value of j? is, according to Art. (54), given by 
 
 Ul,^i + p.+w^+l,'), 
 
 and as m is due to the disturbing action of the sun, we see 
 that the mean angular velocity is less, and therefore the mean 
 periodic time greater than if there were no disturbance. 
 
 Elliptic Inequality or Equation of the Centre. 
 
 73. We shall next consider the effect of the first three 
 terms together : the effect of the second alone, as a correction 
 oipt^ will be discussed in the Historical Chapter, Art. (115). 
 
 • The accurate value was 27d. 7h. 43iii. 11'261b. in the year 1801. See 
 Art. (108). 
 
 K 
 
66 LUNAR THEORY. 
 
 6 =j)t + 2e sin {cpt — a) + le" sin 2 (cpt — o), 
 
 which may be written 
 
 0=pH2e sin [pt- {«+(!- c)p<}]4|e'' sin2[j)<- {a+ (1 -c)p<)]- 
 
 £ut the connexion between the longtitude and the time in 
 an ellipse described about a centre of force in the focus is, 
 Art. (13), to the second order of small quantities : 
 
 6 — nt + 2e sin (nt - a') + |e° sin 2 (nt - a'), 
 
 where n is the mean motion, e the eccentricity, and a' the 
 longitude of the apse.* 
 
 Hence, the terms we are now considering indicate motion 
 in an ellipse ; the mean motion being p, the eccentricity e, 
 and the longitude of the apse a + (1 — c) pt; that is, the 
 apse is not stationary but has a progressive motion in longi- 
 tude, uniform, and equal to (1 — c)p. 
 
 74. The two terms 2c sin (cpt — a) + fe* sin 2 (cpt — a) con- 
 stitute the elliptic inequality, and their effect may be further 
 illustrated by means of a diagram. 
 
 Let the full line AMB re- 
 present the moon's orbit about 
 the earth E, when the time t 
 commences, that is, when the 
 moon's mean place is in the prime 
 radius .ET, from which the lon- 
 gitudes are reckoned. 
 
 The angle TEA, the longi- 
 tude of the apse, is then a. At the time t, when the moon's 
 mean longitude is rEM=pt, the apse line will have moved 
 
 * The epoch < which appears in the expression of Art. (13) is here omitted • 
 a proper assumpUon for the origin of (, as explained in Art. (36), enabling xa 
 to avoid the e. 
 
MOTION OF THE APSE. 67 
 
 in the same direction through the angle AEA' = (1 — c) TEM, 
 and the orbit will have taken the position indicated by the 
 dotted ellipse. The true place of the moon in this orbit, 
 so far as these two terms are concerned, will be m, where 
 
 MEm = 2e sin [cpt - a) + f e' sin 2 {cpt - a) 
 = 2e sin A' EM + fe' sin 2 A' EM 
 = 2e sin^'^Jf (1 + f e cos^'^Jf ) ; 
 
 which, since e is about ^, is positive from perigee to apogee, 
 and therefore the true place before the mean; and the 
 contrary from apogee to perigee: at the apses the places 
 will coincide. 
 
 75. The angular velocity of the apse is (1 — c) p, or, if 
 for c we put the value found in Art. (52), the velocity will 
 be fwi'p. Hence, while the moon describes 360°, the apse 
 should describe f >n'.360° = If ° nearly, m being about ^j. 
 
 But Hipparchus had found, and all modern observations 
 confirm his result, that the motion of the apse is about 3° in 
 each revolution of the moon. See Art. (118). 
 
 This difference arises from our value of c not being repre- 
 sented with sufficient accuracy by 1 — fm*. 
 
 Newton himself was aware of this apparent discrepancy 
 between his theory and observation ; and we are led, by his 
 own expressions (Scholium to Prop. 35, lib. III. in the first 
 edition of the Principia), to conclude that he had got over 
 the difficulty. This is rendered highly probable when we 
 consider that he had solved a somewhat similar problem 
 in the case of the node ; but he has nowhere given a state- 
 ment of his method : and Clairaut, to whom we are indebted 
 for the solution, was on the point of publishing a new hy- 
 pothesis of the laws of attraction, in order to account for 
 it, when it occurred to him to carry the approximations to 
 
68 LUNAR THEORY. 
 
 the third order, and he there found the next term in the 
 value of c nearly as considerable as the one already obtained. 
 See Appendix. The value is 
 
 .-. 1 - c = f m' + V5* Tfi^ = f »i' (1 + V") ; 
 
 .-. (1 - c) 360°= (1 4 //?) (value found previously) 
 
 = 2f ° nearly, 
 
 thus reconciling theory and observation, and removing what 
 had proved a great stumbling-block in the way of all as- 
 tronomers.* 
 
 When the value of c is carried to higher orders of ap- 
 proximation, the most perfect agreement is obtained. 
 
 The motion of the apse line is considered by Newton 
 in his Frincipia^ lib. I., Prop. 66, Cor. 7. 
 
 Evection. 
 
 76. The next term + iJOTesin{(2 - 2)n - c)p<- 2/8 + a} 
 in the value of Q has been named the Evection. We shall 
 consider its eflPect in two different ways. 
 
 First, by itself, as forming a correction on pt, 
 
 e =zpt +^me sin {(2 - 2m-c)pt-20 + a.}. 
 Let J = pt = moon's mean longitude at time f, 
 
 0= mpt + fi =sun'8 , 
 
 a' = (l - c) 2?^ + a = mean longitude of apse , 
 
 then 
 
 =pt + ^me sin [2\pt- {mpt + /3)} - [pt -{1- c)pt + a]] 
 =pt + i/OTfi sin {2 (J - o) - (5 - a')}. 
 
 * See Dr. Whewell's Brulgeicaiei" Treatise, 
 
EVECTION. 69 
 
 The eflFect of this term will therefore be as follows : 
 In syzygies 
 
 6 =pt — i^me sin (S - a') ; 
 
 or the true place of the moon will be before or behind the 
 mean, according as the moon, at the same time, is between 
 apogee and perigee or between perigee and apogee. 
 In quadratures 
 
 6 =pt + ^me sin (]) - a'), 
 
 and the circumstances will be exactly reversed. 
 
 In both cases, the correction will vanish when the apse 
 happens to be in syzygy or quadrature with regard to the 
 sun at the same time as the moon. 
 
 In intermediate positions, the nature of the correction 
 is more complex, but it will always vanish when the sun 
 is at the middle point between the moon and the apse, 
 or when distant 90° or 180° from that point; for if 
 
 O = ^i^ - r.90°, where j- = 0, + 1, or 2, 
 
 sin [2 {D - O) - (5 - a')] = sin (J + a' - 2©) 
 = sin r. 180° 
 = 0. 
 
 77. The other and more usual method of considering 
 the effect of this term is in combination with the two terms 
 of the elliptic inequality, as follows : 
 
 To determine the change in the position of the apse and in 
 the eccentricity of the moon's orbit produced by the evection. 
 
 Taking the elliptic inequality and the evection together, 
 we have 
 
 6 =pt + 2e sin (cpt -a) + ^e' sin 2 {cpt - a) 
 
 + ^me sin {(2 - 2m -c)pt-2^-\- a}. 
 
70 LUNAR THEORY. 
 
 Let a! be the longitude of the apse at time t on supposition 
 of uniform progression, 
 
 O sun ; 
 
 whence a' = {l—c)pt + a, 
 
 O = mpt + fi. 
 And the above may be written 
 
 6 =^pt + 2e sin [cpt — a) + |e' sin 2 [cpt — a) 
 
 + ^-^me sin {cpt - a + 2 (a - o)} ; 
 and the second and fourth terms may be combined into one, 
 
 2^sin(cp<-a+8), 
 if we assume Ecoah = e + ^^tne cos2 (a — o), 
 and ^sinS= '^nze sin2 (a — ©) ; 
 
 whence tan S and E may be found ; and approximately, 
 E=e{l + i^OT cos2 (a - ©)}, 
 S = ^m sin2 [a! — q). 
 
 The term |e' sin 2 (cpt— a) may also, therefore, to the 
 second order, be expressed by 
 
 fj;'''sin2{cp«-a + Sj, 
 
 and the longitude becomes 
 
 d =pt + 2E sin [cpt - a + 3) + \E' sin 2 [cpt - a + S), 
 
 or e=pt + 2Eam {pt - a' + S) + \E' sin2 (jj« - a' + S) ; 
 
 but the last two terms constitute elliptic inequality in an orbit 
 whose eccentricity is E and longitude of the apse a' — S; 
 therefore the evection, taken in conjunction with elliptic 
 inequality, has the effect of rendering the eccentricity of the 
 moon's orbit variable, increasing it by ^me when the apse- 
 line is in syzygy, and diminishing it by the same quantity 
 when the apse-line is in quadrature ; the general expression 
 for the increment being ^ine cos 2 (a' — o). 
 
VARIATION. 71 
 
 And another effect of this terra is, to diminish the longitude 
 of the apse, calculated on the supposition of its uniform 
 progression, by the quantity S = ^ms'm2{a' - O); so that 
 the apse is behind its mean place when in the first and third 
 quadrant in advance of the sun, and before its mean place 
 in the second and fourth. 
 
 The cycle of these changes will evidently be completed 
 in the period of half a revolution of the sun with respect 
 to the apse, or in about -j^ of a year.* 
 
 78. The period of the evection itself, considered inde- 
 pendently of its effect on the orbit, is the time in which 
 the argument (2 — 2m - c)pt - 2/S + a will increase by 27r. 
 
 Therefore period of evection 
 
 277 mean sidereal month 
 
 {2 — 2m — c)p '2 — 2m — c 
 
 mean sidereal month 27j days 
 
 , nearly. 
 
 1-2to + |»i' 1-^ 
 
 = 31^ days, nearly.f 
 
 Newton has considered the evection, so far as it arises 
 from the central disturbing force, in Prop. 66, Cor. 9, of 
 the Principia. 
 
 Variation. 
 79. To explain the physical meaning of the term 
 Vm'sin{(2-277i)^i-2^}, 
 
 * The change of eccentricity and the variation in the motion of the apse 
 follow the same law as the abscissa and ordinate of an ellipse referred to 
 it8 centre : for if £ — e = x and S = y, then 
 
 x^ y' _ 
 
 i The accurate value is 31 ■8119 days. 
 
72 LUNAR THEORY. 
 
 in the expression for the moon's longitude, 
 
 e =^t + Vr/j' sin { (2 - 2Tn)pt - 2/9}. 
 Let ]) represent the moon's mean longitude at time i, 
 
 O BUU'S 
 
 therefore J) =pt, 
 
 O = mpt + j8 ; 
 
 and the value of 6 becomes 
 
 e = p<+ Vjn'8in2(])-o), 
 
 ■which shews that from syzygy to quadrature the moon's 
 true place is before the mean, and behind it from quad- 
 rature to syzygy; the maximum dijBFerence being ^jw* in 
 the octants. 
 
 The angular velocity of the moon, so far as this term 
 is concerned, is 
 
 ■^=p+ V {I - m) m'p cos2 {]i - O), 
 
 =p {l+^m' cos2 (J - o)}, nearly, 
 
 ■which exceeds the mean angular velocity p at syzygiea, is 
 equal to it in the octants, and less in the quadratures. 
 
 This inequality has been called the Variation, its period 
 is the time in which the argument (2 — 2m) pt — 2y3 ■will 
 increase by 27r ; 
 
 ... ... 2'jr mean s'ynodical month 
 
 .-. period of variation = 7- — -— — = = — 
 
 '^ (2 - 2m) p 2 
 
 = 14f days, nearly.* 
 
 80. The quantity y m' is only the first term of an endless 
 series -which constitutes the coeflScient of the variation, the 
 other terms being obtained by carrying the approximation 
 
 * The accurate value is 14 • 765294 days. 
 
ANNUAL EQUATION. 73 
 
 to a higher order. It is then found that the next term in 
 the coefficient is f fm', which is about -j^j- of the first term ; 
 and as there are several other important terms, it is only 
 by carrying the approximation to a higher order (the 5th 
 at least) that the value of this coefficient can be obtained 
 with sufficient accuracy from theory. In fact, y«i' would 
 give a coefficient of 2G' 27" only ; whereas the accurate value 
 is found to be 39' 30". 
 
 The same remark applies also to the coefficients of all 
 the other terms. 
 
 81. As far as terms of the second order, the coefficient 
 of the variation is Independent of e the eccentricity, and k 
 the inclination of the orbit. It would therefore be the same 
 in an orbit originally circular, whose plane coincided with 
 the plane of the ecliptic: it is thus that Newton has con- 
 sidered it. Princip. prop. 66, cor. 3, 4, and 5. 
 
 Annual Equation. 
 
 82. To e3q>lain the physical meaning of the term 
 
 — 3nie' sin [mpt + /8 - f ) 
 in ike expression for the moon's longitude. 
 6=pt- 3me' sin {mpt + y3 - f ), 
 =pt — 3me' sin (longitude of sun - longitude of sun's perigee), 
 =zpt - 3me' sin (sun's anomaly). 
 Hence, while the sun moves from his perigee to his 
 apogee, the true place of the moon will be behind the mean ; 
 and from apogee to perigee, before it. The period being an 
 anomalistic year, the efiect is called Annual Equation. 
 Differentiating we get 
 
 -J —p {1 - 3mV cos (sun's anomaly)}. 
 
74 LUNAR THEORY. 
 
 Hence, so far as this inequality is concerned, the moon's 
 angular velocity is least when the sun is in perigee, that 
 is at present about the 1st of January, and greatest when 
 the sun is in apogee, or about the 1st of July. 
 
 The annual equation is, to this order, independent of the 
 eccentricity and inclination of the moon's orbit, and therefore, 
 like the variation, would be the same in an orbit originally 
 circular. Vide Newton, Principia, prop. 66, cor. 6. 
 
 Eeduction. 
 
 83. Before considering the eflFect of the term 
 
 W 
 --sin2(^p<-7), 
 
 which, as we shall see Art. (89), is very nearly equal to 
 the difference between the longitude in the orbit and the 
 longitude in the ecliptic, it will be convenient to examine 
 the expression for the latitude of the moon, and to see how 
 the motion of the node is connected with the value of g. 
 
 LATITUDE OF THE MOOX. 
 
 84. The expression found for the tangent of the latitude,* 
 Art. (51), is 
 
 s = i sin [g0 - 7) 4 ^mk sin {{2- 2m-g]d-2^ + 7]. 
 
 If we reject all small terms, we have 
 
 s = 0, 
 
 or the orbit of the moon coinciding with the ecliptic, which 
 is a first rough approximation to its true position. 
 
 * This expression for the tangent of the latitnde is more convenient than 
 that which gives it in terms of the mean longitude, Art. (57), on account of 
 the less number of terms involved. See Pontecoulant, vol. iv. p. C30, 
 
LATITDDE OF THE MOON. 75 
 
 85. Taking the first term of the expansion 
 
 we may write it 
 
 s = k3m[d-{y-{g-l)e}]. 
 
 Let TNm be the ecliptic, N the 
 moon's node when her true longitade 
 is zero, and let M be the position 
 of the moon at time t, m her place 
 referred to the ecliptic ; 
 
 therefore TN^y, Tm = 6, ta,u.Mm = s. 
 
 Take NN' ={g — l)0 in a retrograde directionj and join 
 IfN' by an arc of great circle ; 
 
 then tan Mm = tan MN'm sin N'm. 
 
 or s = iasi MN'm smie-{y-{g- 1)6]]; 
 
 which, compared with the value of s given above, shews that 
 MN'm = tan''A is constant, and therefore the term k sin {g0—y) 
 indicates that the moon moves in an orbit inclined at an 
 angle tasi^k to the ecUptic, and whose node regredes along 
 
 the ecliptic with the velocity {g-l)-jr, or with a mean 
 
 velocity {g — l)p- 
 
 86. Hence the period of a revolation of the nodes 
 
 271- one sidereal month 
 
 but, from Art. (51), the value of ^ = 1 + f»i' ; 
 
 . , , , . , J one sidereal month 
 therefore period of revolution or nodes = ^—^ 
 
 = 6511 days, nearly. 
 
 This will, for the same reason as in the case of the apse, 
 Art. (75), be modified when we carry the approximation 
 
76 LUNAR THEORY. 
 
 to a higher degree ; this value of g is, however, much more 
 accurate than the corresponding value of c, for the third 
 term of g is small ; the value to the third order being (see 
 Appendix, Art. 102) 
 
 g=l + lm'--i^m% 
 
 , , . 1 ,. , . ,. , 1 one sidereal month 
 
 and the period of revolution ot the nodes = 5 — §— ; — ; — 
 
 ^ fin (1 — |»n) 
 
 = 6705 days, nearly. 
 
 This is not far from the accurate value as given by 
 observation, and when the approximation to the value of g 
 is carried to a higher order, the agreement is nearly perfect. 
 
 The true value is 6793'39 days, that ia about 18 yrs. 7 mo. 
 
 Evection in Latitude. 
 
 87. To explain the variation of the inclination and the 
 irregularity in the motion of the node ea^essed hy the term 
 
 + fm4sin {(2- 2m-^)^-2/3 + 7}. 
 
 This term, as a correction on the preceding, is analogous 
 to the evection as a correction on the elliptic inequality. 
 Taking the two terms together, 
 
 s = A sin {gd - 7) + %mk sin {(2 -2m-g)0-2^ + 7}. 
 Let ]) = longitude of moon = 9, 
 
 0= sun =m6 + 0, 
 
 £3= node =7 — (^^-l)^; 
 
 therefore s = A sin (]) - §^ ) + fwiA- sin {J - g^ - 2 (O - g^)|. 
 Now these two terms may be combined into one, 
 s = KsinO- Q -A), 
 if K COS A = k + ^mk coa2 (O - a), 
 
 KsmA= |mAsin2(0- G), 
 
REDUCTION IN LONGITUDE, 77 
 
 whcDce A and K may be found ; and, approximately, 
 X=^{l + fjnco82(0-g^)}, 
 A = §OTsin2(0- Q), 
 but the equation 
 
 s = Kain('])- Q-A) 
 represents motion in an orbit inclined at an angle tan"'^ to 
 the ecliptic, and the longitude of whose node is Q + A. 
 
 This term has therefore the following effects : 
 
 1st. The inclination of the moon's orbit is variable, its 
 tangent increases by ^mk when the nodes are in syzygies, 
 relatively to the sun, and decreases by the same quantity 
 when they are in quadrature ; the general expression for 
 the increase being ^mk cos2(0 — £3). 
 
 2nd. The longitude of the node, calculated on supposition 
 of a uniform regression, is increased by A = fm 8in2 (q — S^), 
 so that the node is before its mean place while in the first 
 or third quadrant in front of the sun, and behind it in the 
 second and fourth. Principia, book iii., props. 33 and 35. 
 
 The cycle of these changes will be completed in the 
 period of half a revolution of the sun with respect to the 
 node, that is, in 173-21 days, not quite half-a-year. 
 
 88. The tangent of the latitude has here been obtained ; 
 if we wish to have the latitude itself it will be given by the 
 formula 
 
 latitude = s-^s*+ ^s" - &c. 
 which, to the degree of approximation adopted, will clearly 
 be the same as s. 
 
 Beduction. 
 
 89. We may now consider the term which we had 
 neglected (Art. 83) in the expression for the longitude, namely, 
 
 — Ik^ siQ2 (gpt — y). 
 
78 
 
 LUNAR THEORY. 
 
 Let N be the position of the node when the moon's 
 longitude is d, M the place of the 
 moon, m the place referred to the 
 ecliptic. 
 
 Therefore Tm = 6, 
 
 Nm =g6 — y, 
 tan N=k. 
 
 The right-angled spherical triangle NMm gives 
 tan^w 
 
 therefore 
 
 or 
 
 1 
 
 cos N= =r=y. , 
 
 tan NM ' 
 cos N tan NM - tan Nm 
 
 1 + cos^ tan NM-^ tan Nm ' 
 
 ,N _ a\n (NM-N m) 
 2 ~ smiNM+Nm)' 
 
 or, since both N and NM— Nm are small, 
 
 tan'iV NM- Nm . , , 
 
 therefore NM-Nm=lk' 8in2(^^— 7)= JA' sin 2(9^*— 7), nearly. 
 
 Hence this term, which is called the reduction, is approxii 
 mately the difference between the longitude in the orbit and 
 the longitude in the ecliptic. 
 
 RADIUS VECTOR. 
 
 90. To explain the physical meaning of the terms in the 
 value of u. 
 
 We shall, for the explanation, make use of the formula 
 which gives the value of u in terms of the true longitude, 
 Art. (52). 
 
TERMS IN RADIUS VECTOR. 79 
 
 Neglecting the periodical terms, we have for the mean 
 value 
 
 The term — ^m', which is a consequence of the disturbing 
 effect of the sun, shews that the mean value of the moon's 
 radius vector, and therefore the orbit itself, is larger than if 
 there were no disturbance. 
 
 Elliptic Inequality. 
 
 91. To explain the effect of the term of the first order, 
 
 u = a {l + ecos(c0 — a)}, 
 
 = a [1 + e cose-{a + (l-cj^j]. 
 
 This is the elliptic inequality, and Indicates motion in an 
 ellipse whose eccentricity is e and longitude of the apse 
 a + (l— c)^; and the same conclusion Is drawn with respect 
 to the motion of the apse as in Art. (73). 
 
 Evection. 
 
 92. To explain the physical meaning of the term 
 
 ^mea cos {(2 - 2?w - c) - 2/3 + a}. 
 
 This, as in the case of the corresponding term in the longi- 
 tude, is best considered in connection with the elliptic 
 inequality, and exactly the same results will follow. 
 
 Thus, calling J), O, and a! the true longitudes of the moon, 
 sun, and apse (the last calculated on supposition of uniform 
 motion), these two terms may be written, 
 
 M = a [1 + e cos(}) - a')- ^me cos {5 - a' + 2 (a' - O)}] 
 = a[l + ^cos(])-a' + 8)]; 
 where ^cos8 = e+ '/OTecos2(a'- o), 
 
 JE^ sin S = ^me sin2 (a - o). 
 These are identical with the equations of Art. (77). 
 
80 LUNAR THEORY. 
 
 Variation. 
 
 93. To explain the effect of the term m'a cos {(2-2m) 0- 2/S}, 
 
 M = a[l + OT*co3l(2- 2m)^-2/3}] 
 = o[l + »n'co3 2(])-Q)]. 
 
 As far as this term is concerned, the moon's orbit would 
 be an oval having its longest diameter in quadratures and 
 least in sjzygies, Principia, lib. I. prop. 66, cor. 4. 
 
 The ratio of the axes of the oval orbit will be 
 
 ^ = §§ nearly, m being •0748. 
 
 See Principia, lib. III. prop. 28. 
 
 Meduction. 
 
 94. The last important periodical term in the value of u is 
 
 - — CQ32(g0-y). 
 
 This term expresses approximately the variation in the 
 difference between the values of u in the orbit and in the 
 ecliptic. 
 
 For if u^ be the reciprocal of the value of the radius 
 vector in the orbit, 
 
 Mj = M cos (latitude), 
 
 therefore u — u^ = ^us' = ^ak' sin' (g9 - 7) 
 
 c=laK'-^ak'cos2(ge-y) 
 
 = const. — ^a/c' cos 2 (gd — 7). 
 
 Taking this result in connexion with that of Art. (89) 
 we see that the reduction, so far as periodical terms to the 
 
PERIODIC TIME OF THE MOON. 81 
 
 "second order are concerned, is simply a geometrical con- 
 sequence of the inclination of the orbit ; and that, if we 
 measured the longitude and the radius vector along the 
 orbit instead of taking their projection on the plane of the 
 ecliptic, these periodical terms would not appear. 
 
 95. There are no other terms of the second order in the 
 value of u. The annual equation, which, in the longitude, 
 is of the second order, is only of the third order in the 
 radius vector. 
 
 Periodic time of the Moon, 
 
 96. We have seen. Art. (72), that the periodic time of 
 the moon is greater than if there were no disturbing force ; 
 but this refers to the mean periodic time established on an 
 interval of a great number of years, so that the circular 
 functions in the expression are then extremely small com- 
 pared with the quantity pt which has uniformly increased. 
 
 When, however, we consider only a few revolutions, these 
 terms may not all be neglected. The elliptic inequality and 
 the evection go through their values in about a month, the 
 variation and reduction in about half-a-month ; their effects, 
 therefore, on the length of the period can scarcely be con- 
 sidered, as they will increase one portion and then decrease 
 another of the same month. 
 
 But the annual equation takes one year to go through its 
 cycle, and, during this time, the moon has described thirteen 
 revolutions; hence, fluctuations may, and, as we shall now 
 shew, do take place in the lengths of the sidereal months 
 during the year. 
 
 We have, considering only the annual equation. Art. (82), 
 
 pl=^e + 3me sin (md + /3-^). 
 
 JI 
 
82 LUNAR THEORY. 
 
 Let T be the length of the period : then, when Q i* 
 increased by 27r, t becomes <+ Z"; 
 
 therefore ^ (< + T) = 2ir + ^ + 3me' sin (2ffi7r + m^ + /3 - ?), 
 whence ^7'=27r+ ^me sinraTT cos(7n7r + md-'c ^ — f ) ;. 
 
 therefore T= mean period H sin?w7r cos(o - Oi 
 
 Jr 
 
 where © =m^ + y9+ »i'7r = sun's longitude at the beginning, 
 of the month + mir 
 = sun's longitude at the middle of the month. 
 Hence Twill be longest when o - f=0, 
 and shortest when — f = tt ; 
 
 or T will be longest when the sun at the middle of the month. 
 is in perigee, and shortest when in apogee ; but, at present,, 
 the sun is In perigee about the 1st of January, and in apogee 
 about the 1st of July; therefore, owing to annual equation^ 
 the winter months will be longer than the summer months, 
 the difference between a sidereal month in January and July, 
 from this cause, being about 20 minutes. 
 
 97. All the inequalities or equations, which our expres- 
 sions contain, have thus received a physical interpretation. 
 They were the only ones known before Newton had estab- 
 lished his theory, but the necessity for such corrections was 
 fully recognized, and the values of the coefficients had already 
 been pretty accurately determined ; still, with the exception 
 of the reduction, which is geometrically necessary, they were 
 corrections empirically made, and it was scarcely to be ex- 
 pected that any but the larger inequalities, viz. those of the 
 first and second orders which we have here discussed, could 
 be detected by observation : we find, however, that three 
 others, have, since Newton's time, been Indicated by obser- 
 vation before theory had explained their cause. These are — 
 
INEQUALITIES FOUND BY OBSEETATION. 83 
 
 Ihe secular acceleration, discovered by Halley ; an inequality, 
 found by Mayer, in the longitude of the moon, and of which 
 the longitude of the ascending node is the argument; and 
 finally an inequality discovered by Biirg, which has only of 
 late years obtained a solution. For a further account of 
 these, as also of some other inequalities which theory has 
 made known, see Appendix, Arts. (108) to (112). 
 
( 8-i ) 
 
 CHAPTER VII. 
 
 APPENDIX. 
 
 In tins chapter will be found collected a few propositions 
 intimately connected with the results or the processes of the 
 Lunar Theory as explained in the previous pages. Refer- 
 ence has been made to some of them in the course of the 
 work, and the interest and importance of the others are 
 sufficient to justify their introduction here. 
 
 98. The moon is retained in her orbit hy the force of 
 gravity^ that is, by the same force which acts on bodies at 
 the surface of the earth. 
 
 The proof of this is merely a numerical verification ; the 
 data required from observation are, 
 
 the space fallen through from rest in i" by bodies at the 
 
 earth's surface = 161 feet, 
 
 the radius of the earth = 4000 miles, 
 
 the periodic time of the moon = 27J days, 
 
 the distance of the moon from the earth's centre=60x4000 miles. 
 
 The force of the earth's attraction oc — -. — ^ . Therefore, the 
 
 (dist.)" ' 
 
 space fallen through in 1" at distance of moon by a body 
 
 1 fi'1 
 moving from rest under the earth's action = -— ^ feet 
 
 = -00447 feet. 
 
J[OON S ORBIT CUN'CAVK. 85 
 
 But the moon in one second describes an ancrle — ; =ai, 
 
 •= 27J.24.60' ' 
 during which the approach to the earth 
 
 = 60 X 4000 X 5280 (vers, w) feet 
 60 X 4000 X 5280 x 2ir' 
 
 (27i/.(24y.(60/ 
 = -00448 feet. 
 
 feet 
 
 Therefore, tlie space through which tlie moon is deflected 
 in one second from her straight path, is just the quantity- 
 through which she would fall towards the earth, supposing 
 her to be subject to the earth's attraction; and we may, 
 therefore, conclude that she is retained in her orbit by the 
 force of gravity. 
 
 When first Newton, in 1666, attempted to verify this 
 result, he found a difference between the two values equal 
 to one-sixth of the less: the reason of his failure was the 
 incorrect measures of the earth, which he made use of in 
 his computation; and it was not till about 16 years later 
 that he was led to the true result, by using the more 
 correct value of the earth's radius obtained by Picart. 
 
 Principia, lib. III., prop. 4. 
 
 99. The moon's orbit is everywhere concave to the sun. 
 
 Let S, E, and M be the centres of the sun, earth, and 
 
 moon. We ought first to apply ^ yn 
 
 to each body forces equal and ■^- / 
 
 opposite to those which act on •"■ 
 
 the sun in order to bring him to rest. These forces are, 
 however, so small that we may neglect them, and we shall 
 consider the moon to be moving about the sun fixed, and to 
 be disturbed by the earth alone. 
 
86 LDNAE THEORY. 
 
 f 
 
 The forces on M are, therefore, „.., in MS, 
 
 ^"^^m-^ '°^^- 
 
 This last must be resolved into two, one in MS, the other 
 perpendicular to it. 
 
 Therefore, the whole central force on the moon in MS 
 
 m' E -, 
 
 and the proposition will be proved if we shew that this force 
 is always positive. Now, Art. (25), 
 
 . ,. . , 2-7r. SG^ 2ir.SM" 
 periodic time ot suu = rr — = ^r — nearly, 
 
 iir.EM^ 
 and moon 
 
 {E+M)i' 
 therefore -g^, = ^^-g -^^ nearly ; 
 
 therefore -g^. > j J^ -g^„ , 
 
 m , SM E ,,. E 
 
 SM" ^^^ EM ' EM' '^^ EM" ' 
 
 therefore -^^ - -^^^ is positive : 
 
 but this is the value of the central force corresponding to 
 «osif=— 1, and is therefore its least value. Hence the 
 central force always tends to the sun, and the path is always 
 concave. 
 
 At new moon the force with which the moon tends to the 
 sun is, therefore, greater than that with which she tends to 
 the earth: the earth being itself in motion in the same 
 
CENTRAL DISTURBING FORCE. 87 
 
 direction, and, at that instant, with greater velocity, will 
 easily explain how, notwithstanding this, the moon still 
 revolves about it.* 
 
 Central and Tangential Disturbing Forces. 
 
 100. We have hitherto considered the effects of the 
 
 central and tangential disturbing forces in combination ; but 
 
 it will be interesting to determine to which of them the 
 several inequalities principally owe their existence. 
 
 (1) To determine the effect of the central disturbing force. 
 
 Make T=0; 
 
 L r d-'u P „ 
 
 therefore j7,2 + m - rr-i = ", 
 
 av ti u 
 
 P 
 
 or substituting for 77-^ from Art. (52), 
 
 7 J a-lk'-W + ^m^e cos c^, - fm' cos (2 - 2w2) ^, 
 
 j^"+M=a +|7n'ecos(2-2m-c)^. 
 
 '^ I +f/l'cos2^^,; 
 
 ^erefore, 
 
 n _ |A'_^m'4e cos(c0- a) + iW cos{(2 - 2m)5-2/3} 
 u=a\ ,H-^gjnecosl(2-2»n-c)0-2y3 + a} 
 
 If we compare this with the value of u found Art. (52")j we 
 see that the elliptic Inequality and the reduction are due 
 to the central or radial force, as also one-half of the varia- 
 tion and three-tenths of the evection. 
 
 It would perhaps be proper to separate the absolute cen- 
 tral force from the central disturbing force; the terms due 
 to the latter are those which contain m ; therefore, the elliptic 
 
 * See the Author's Astronomy, p. 233. 
 
88 LUNAR THEORY. 
 
 inequality and the reduction are the effects of the former, 
 except that in the elliptic inequality the introduction of c, or 
 the motion of the apse, is due to the disturbing force. 
 
 (2) To determine the effect of the tangential disturbing force. 
 
 Let the central disturbing force be zero ; 
 
 then ,2-2 = 7j (1 — f «'■') = a, neglecting the inclination, and, 
 
 substituting for ^^,-3 . ^^ and for 2 [-^^ + «J J^, dd their 
 values from Art. (52), the equation in u becomes 
 
 d'u 
 
 —J- + M = a { 1 - ?^ni' cos (2 - 2m) 6^ + 2J m'e cos (2 - 2m - c) 0^} ; 
 
 , _ j 1 + e cos (c^ - a) 4 ^m' cos {(2 - 2m) d - 2^} 
 
 whence M_a| +|^^eeos|(2-2m-c) ^-2/3 + a}. 
 
 We have here the remaining half of the variation and the 
 remaining seven-tenths of the evection as the effects of the 
 tangential disturbance. Also c = l, or, to the second order, 
 the tangential force has no effect on the motion of the apse. 
 
 101. The separate effects of the central and tangential 
 forces in producing the inequalities in the longitude may be 
 traced in a similar manner. We shall leave this as an 
 exercise for the student. The annual equation will be found 
 to be due to the central force. The other inequalities will 
 be divided as in the radius vector; except as regards the 
 variation, of which four-elevenths are due to the central 
 and the remaining seven-elevenths to the tangential force. 
 
 To calculate the value of g to the third order. 
 
 102. We must here make use of the results which the 
 approximations to the second order have furnished ; but as 
 the value of g is determined by that terra of the differential 
 equation whose argument is gd — y, we need only consider 
 
C AND g TO THE THIRD ORDER. 89 
 
 those terms which by their combinations will lead to It with- 
 out rising to a higher order than the fourth. 
 From Arts. (51), (31), we obtain 
 
 8 = k smgd^ + ^mk sin (2 — 2m —g) 0^, 
 
 Ps- S 
 ,, 3 = - |9k"« {1 + cos (2 - 2m) 6^}, 
 
 Hi Hi 
 
 ^, = -3m'sin(2-2m)e.. 
 
 From these we obtain 
 
 Ps— 8 
 
 -j^ =...-lm'k{l- fjm) SingO^ 
 
 -j^,. J =...-|OT'A;8in(2-2»i)e,{cos5f^, + §»ico3(2-2m-^) 6^} 
 a u do 
 
 =...--^>»A8in^^., 
 
 d's 
 
 . 1- s = 0, to the second order : 
 
 dff' ' 
 
 therefore 2 (^ + «) J^. ^^ = 0, to the fourth order. 
 
 Therefore the equation in s becomes, so far as these terms 
 are concerned, 
 
 ^ + s = (- fm'A + ^^m'k + ^\m'k) slngd, ; 
 
 assume s = A;sin^^„ 
 
 therefore A (1 - /) = - fm'A + i\ m% 
 
 g = l + lm'--i^m\ 
 
 To find the value of c to the third order. 
 103. Proceeding in a similar manner from the equations. 
 Arts. (52), (31), we get 
 M=a{l+ecosc^,+m'cos(2-2»i)^,+ Vwecos(2-2»w-c)5,}, 
 
 N 
 
90 LUNAR THEORY. 
 
 ^- =a- im' ^,{1 + 3 cos (2 - 2m) 6,}, 
 
 we obtain 
 
 p 
 
 rr-2 = a- ^ni'a [I - 3e cosc0,...- ^jne cos (2 - 2ot - c) 5J 
 
 X {1 + 3 cos (2 -2m) 0J 
 = a + ^m'tt {3e + VV'"^} cosc^,, 
 
 yf-, =-f wj' f l-4e cosc^,-4nt'' cos(2-2m) B-^^me cos(2-2»i-c)0J 
 h u 
 
 X sin (2 - 2m) 0, 
 
 = -f wi' {sin (2 - 2 to) ^, - 2e sin (2 - 2m - c) ^, - '^ me sin cO^} , 
 ^=-a{e8ince,+2m'sin(2-2m)e,+ i^mesin(2-2m-c)^J; 
 
 therefore j^,. 33=... ||m'ae cos c0„ 
 
 the other term is of the fifth order. 
 
 /■ T 
 
 I d^=+|m' cos (2-2m) ^j-3m'e cos (2-2m-c)^,- V^'e cosc0„ 
 
 jh u 
 
 f^^ + y\ =« {1...- 3m' co8(2-2m) 0,+ V»»''ecos(2-2m-c)^,; 
 
 5^' "^ ") Ia^ <^^ = - V'^'ofi COSC0,, 
 and the equation in u becomes 
 
 5+ M = o {I + (fm'e + W»'e- fKe+ V^'e) cosc0, +,..} 
 = a{l + (|m'e+ ^^^ni'e) cosc^, + ...}. 
 Assume M = a(l + e cosed, +...) 5 
 
 therefore ae(l-c') = (fmV+ V^m'e)a, 
 
 whence c = 1 - |m' - V^m^ 
 
PARALLACTIC INEQUALITY. 91 
 
 104. Hence, to the third order of approximation, 
 
 mean motion of apse _ 1 — c _ f »i' + W*" ^ + ^^^ 
 mean motion of node ~ g — l~ fm' — /gw' ~ 8 — 3»i ' 
 
 and since m = ^ nearly, we see that the moon's apse 
 progredes nearly twice as fast as the node regredes. 
 
 In the case of one of Jupiter's satellites, the periodic 
 time round Jupiter is only a few of our days, and the 
 periodic time of Jupiter round the sun is 12 of our years, 
 therefore m, the ratio of these periods, is very small. 
 
 Hence, the apse of one of Jupiter's satellites progredes 
 along Jupiter's ecliptic, with pretty nearly the same velocity 
 as the node regredes, assuming these motions to be due to 
 the sun's disturbing force; they are, however, principally 
 due to the oblateness of the planet. 
 
 Parallactic Inequality. 
 
 105. In carrying on the approximations to a higher order, 
 it is foimd that the expressions for the forces contain terms 
 whose argument is the moon's elongation, i.e. the difference 
 of longitude of the sun and moon. These terms appear 
 in P and in T of the fourth order, and since 1 —m, the 
 coefficient of 6, is near unity, the terms will become of the 
 third order, and therefore of considerable importance in the 
 values of u and of t. We shall work out these expressions 
 from the earliest steps ; and we select these terms on account 
 of the peculiar use which has been made of them to determine 
 the sun's parallax, whence they have received the name of 
 parallactic inequality. 
 
 Let us go back to the expressions for the forces 
 Arts. (23), (24). In the first place we may remark that 
 the terms we are about to compute must be independent 
 of s ; because any term into the composition of which s enters 
 
92 
 
 LUNAR THEORY. 
 
 ■will necesBarily have g in the coefficient of the argflment. 
 We may therefore at the outset suppose the orbit to coincide 
 with the ecliptic. We have then 
 
 SM' = r'^ - 2r'MG cos (^ - ff) + M0\ 
 
 SM^ = 7^ l + -^cos(5-e')-f7^+V^7F-cos'(^-^)J 
 1 + ^ cos(0 - 6') + ^^- W+^i cos2(0-^')}l , 
 
 T T J 
 
 We must now substitute these values in 
 
 E±M ,(MG EG\ , ,/ 1 1 \ „ 
 
 so 
 
 P= 
 
 -'"''■' Im-^-M'J'^°(^-^'^' 
 
 and pick out the terms which have for argument {d — 6'). 
 
 T , /MG EG\ ,, .,, . , 
 
 In m I -QT^3 + ^E^s 1 there will be a term 
 
 3m' ^, cos(^-e') = -^ __ co8(^-5'). 
 
 In m'r'(-^jr^— =^ J cos (5-^') we shall have the terms 
 
 ,{,M(?-EG^ {f cos(^-0') + '^ co&{e-e') coi2{e-e% 
 
 m 
 
 which produce 
 
 , [MG'-EG'} 
 
 (1+ .^) cos(e-0') = ^ ^ CT«^°«(^- ^')> 
 
 and finally mV f-oijjs - "cpa ' sin(0- 6') will produce 
 ,MG'-EG\^ ,,. . ,a m oWiV^-iW. ,. .,. 
 
PARALLACTIC INEQUALITY. 93 
 
 Therefore the new terms will be 
 
 Since — (= -^^-^ nearly) is of the second order, these terms 
 
 are of the fourth order. 
 
 106. On the principle of the superposition of small 
 disturbances, we may compute the effects of these terms by 
 themselves as disturbers of the elliptic path. 
 
 T^,= -kS— S-8m{(l-»n)e-/3}, 
 
 AV •* E+Ma '^ ' '' 
 
 u = a{l+e cos{cd — a.)+ }, 
 
 -j-i = — oe sm [ca — 
 do 
 
 a 
 
 therefore 
 
 rp J 
 
 rrj- Ti} = 0, to the fourth order, 
 A u da 
 
 -j^ +M=a, to the second order, 
 d(r 
 
 /) 
 
 Substituting in the differential equation for m, we get 
 
 s+»-<-h -v™-§=i:-co.i(i-»)«-«+-]- 
 
 Assume u = a 1+ '^■^E+Ma c°sl(l~'")^~^l+— ' 
 
94 LUNAR THEORY. 
 
 therefore A = - — jf- r-^ =-\\m\ 
 
 1 — (1— «i) 
 
 therefore u=a 1+ -\%'">- 'W~m~ cos{(l-»n) d-^\\... . 
 
 107. The corresponding term in the value of Q will also 
 be of the third order, 
 
 i4»h + V-|^:-cos{(i-«^)^-^}+ ], 
 
 ! T . . 
 
 lyj— 5 dO is of the fourth order and will not rise in t ; 
 
 therefore 
 
 and^=p< + - V»»;^^^^ 8in{(l-m)p«-/3}+... . 
 
 Mayer was the first who applied this term to the deter- 
 mination of the sun's parallax, by comparing the analytical 
 expression of the coeflBcient with its value as deduced from 
 
 E 
 observation. The values of m and of ^i>, and therefore of 
 
 p — p being pretty accurately known, — will be determined, 
 
 that is, the ratio of the sun's parallax to that of the moon ; 
 but the moon's parallax is well known ; therefore, also, that 
 of the sua can be calculated. The value so obtained by 
 Laplace was 8'6"; but Hansen, by the discussion of a great 
 
SECULAR ACCELERATION. 95 
 
 number of modern observations, has found 8"92", a value 
 ■which agrees very closely with recent determinations by other 
 methods. 
 
 Secular Acceleration, 
 
 108. Halley, about 1693, found, by the comparison of 
 ancient and modern eclipses, that the moon's mean revolution 
 is now performed in a shorter time than at the epoch of the 
 recorded Chaldean and Babylonian eclipses. This pheno- 
 menon, called the secular acceleration of the moon's mean 
 motion, has not yet been fully accounted for. For a long 
 time it was altogether unexplained, but in 1787 Laplace 
 gave what seemed to be a satisfactory and complete account 
 of it. 
 
 The value of p, Art. (54), on which the length of the 
 mean period depends, is found, when the approximation is 
 carried to a higher order, to contain the quantity e the 
 excentricity of the earth's orbit. Now, this excentricity is 
 undergoing a slow but continual change from the action of 
 the planets, and therefore p, as deduced from observations 
 made in diflferent centuries, will have diflferent values. 
 
 The value of p is at present increasing, or the mean 
 motion is being accelerated, and it will continue thus to 
 increase for a period of immense, but not infinite duration ; 
 for, as shewn by Lagrange, the actions of the planets on 
 the excentricity of the earth's orbit will be ultimately 
 reversed, e will cease to diminish and begin to increase, and 
 consequently p will begin to decrease, and the secular 
 acceleration will, so far as this cause is concerned, become a 
 secular retardation. 
 
 It is worthy of remark that the action of the planets on 
 the moon, thus transmitted through the earth's orbit, is more 
 considerable than their direct action. 
 
96 LUNAR THEORY. 
 
 109. In his investigation, Laplace treated the excentricity 
 e' of the earth's orbit as a constant in the differential equa- 
 tions, and only considered its variation in the final result. 
 This of course greatly simplified the work, but ought to have 
 been looked upon only as a first approximation ; yet, strange 
 to say, the result so obtained agreed almost perfectly with 
 that which had been deduced from a comparison of ancient 
 and modern observations. " Malheureusement cette mer- 
 veilleuse concordance 3, laquelle Laplace a dfi peut-etre une 
 partie de I'^clat de sa plus belle d^couverte, est sensiblement 
 altirie par les approximations sulvantes."* 
 
 Some years ago Professor Adams took up the question 
 ah initio without any limitation, and Introduced the variability 
 of e into the differential equations themselves. This strictly 
 correct process gave a value of the secular variation only 
 about half of Laplace's which had agreed so well with 
 observation. 
 
 The variation of excentricity does not therefore account for 
 the whole observed change, and the mean motion of the moon 
 is affected by some other cause or causes at present unknown. 
 Perhaps the remaining acceleration is only apparent, and 
 arises from a gradual retardation of the earth's diurnal motion. 
 The effect would be precisely the same, and such a retar- 
 dation may be due to the action of the tides. Whatever be 
 the cause, the explanation of this unexplained phenomenon is 
 a question worthy of the mathematician's most serious efforts. 
 
 Inequalities depending on the Figure of the Earth. 
 
 110. The earth, not being a perfect sphere, will not at- 
 tract as if the whole of its mass were collected at its centre : 
 hence, some correction must be introduced to take into 
 account this want of sphericity, and some relation must exist 
 
 * FoDtecoulant, Sysleme du Monde. Supplement, 1860, 
 
RECENT DISCOVERIES. 97 
 
 between the oblateness and the disturbance it produces. 
 Laplace in examining its effect found that it satisfactorily 
 explained the introduction of a term in the longitude of the 
 moon, which Mayer had discovered by observation, and the 
 argument of which is the true longitude of the moon's 
 ascending node. 
 
 By a comparison of the observed and theoretical values 
 of the coefficient of this term, we may determine the 
 oblateness of the earth with as great accuracy as by actual 
 measures on the surface. 
 
 111. By pursuing his investigations, with reference to 
 the oblateness, in the expression for the moon's latitude, 
 Laplace found that it would there give rise to a term in 
 which the argument was the true longitude of the moon. 
 
 This term, which was unsuspected before, will also serve 
 to determine the earth's oblateness, and the agreement with 
 the result of the preceding is almost perfect, giving the 
 compression s^j* which is about a mean between the 
 different values obtained by other methods. 
 
 Perturbations due to Venus. 
 
 112. After the expression for the moon's longitude had 
 been obtained by theory, it was found that there was still 
 a slight deviation between her calculated and observed places, 
 and Biirg, who discovered it by a discussion of the observa- 
 tions of Lahire, Flamsteed, Bradley, and Maskelyne, thought 
 it could be represented by an inequality whose period would 
 be 184 years and coefficient 15". This was entirely con- 
 jectural, and though several attempts were made, it was n<k 
 accounted for by theory. 
 
 ♦ Pontecoulant, Systhne da Monde, vol. iv. 
 
98 LUNAR THEORY. 
 
 About 1848, Professor Hansen, of Seeberg, in Gotha, 
 having commenced a revision of the Lunar Theory, found 
 two terms, which had hitherto been neglected, due to the 
 action of Venus. One of them is direct and arises from 
 a 'remarkable numerical relation between the anomalistic 
 ' motions of the moon and the sidereal motions of Venus 
 ' and the earth ; the other is an indirect effect of an inequality 
 'of long period in the motions of Venus and the earth, 
 'which was discovered some yesu-s ago by the Astronomer 
 ' Royal.'* 
 
 The periods of these two inequalities are extremely long, 
 one being 273 and the other 239 years, and their coefficients 
 are respectively 27'4" and 232". ' These are considerable 
 ' quantities in comparison with some of the inequalities already 
 'recognised in the moon's motion, and, when applied, they 
 ' are found to account for the chief, indeed the only 
 'remaining, empirical portion of the moon's motion in 
 ' longitude of any consequence ; so that their discovery may 
 ' be considered as a practical completion of the Lunar Theory, 
 ' at least for the present astronomical age, and as establishing 
 ' the entire dominion of the Newtonian Theory and its 
 ' analytical application over that refractory satellite. 'f 
 
 Motion of the Ecliptic. 
 
 113. We have seen, Art. (14), that our plane of reference 
 is not a fixed plane, but its change of position is so slow 
 that we have been able to neglect it, and it is only when 
 the approximation is carried to a higher order that the 
 necessity arises for taking account of its motion. 
 
 ♦ Report to the Annual General Meeting of the Royal Astionomical Society, 
 Feb. 11, 1848. 
 
 t Address of Sir John Herschel to the Meeting of the Royal Astronomical 
 Society. 
 
MOTION OP THE ECLIPTIC. 99 
 
 It has been found to have an angular velocity, about an 
 axis in its own plane, of 48" in a century, and the correction 
 thUs introduced produces in the latitude of the mooa a term 
 — CO) cos [0 — <f>), 
 
 where w is the angular velocity of the ecliptic, - the angular 
 
 c 
 
 velocity with which the ascending node of the moon's orbit 
 
 recedes from the instantaneous axis about which the ecliptic 
 
 rotates, (j> the longitude of this axis at time t, and the 
 
 longitude of the moon at the same instant. 
 
 Let TAm be the position of the ecliptic at time <, 
 A the point about which it is turning, TA = <f), 
 MN the moon's orbit, Jf the moon, and Mm a perpen- 
 dicular to the ecliptic ; Twt = 6 ; Mm = lat. = j8. 
 t the inclination of the orbit, and N the longi- 
 tude of the node. 
 Let AFN'tfl be the ecliptic after a time S<. 
 Any point whose longitude is h may be considered as 
 moving perpendicularly to the ecliptic with a velocity 
 o) sin{Z/— <^). 
 
 Hence, the point B will move in the direction PN with 
 a velocity <» sin [N— ^). And N' will move along PN' with 
 a velocity w sin [^- ^) coti; 
 
 therefore -^ = o» sin (i^- ^) cott. 
 
 Again, the point of the ecliptic 90° in advance of N 
 will move towards the moon's orbit with a velocity 
 o) sin (90° + N- 0) ; 
 
100 LUNAR THEORY. 
 
 di 
 
 therefore --j- = — <o cos {N— (j>). 
 
 Now, cot 1, ft), and — ^>— ^ = - - may be considered con- 
 
 stant in integrating ; 
 
 therefore SN^ ca> cos (N— <f>) cot i, 
 
 S{=cto sin {N- (f>), 
 and if NM= yjr, we have 
 
 £>t- = .= --. — .co3(N-d>]. 
 
 cos I sin ^ 
 
 Now, sin /S = sin t. sin ■yfr • 
 cos /8 . 8/3 = cos I . sin i^ . Si + sin i. cos yfr . Syfr 
 
 = ca> {costsini/r8in(^- ^) — cosi|rcos(37"— 0)}, 
 but cosi.sini/r=cos;Ssin(5-^) and cosi^=cos/3cos(0— .JT); 
 therefore 8/3 = — cft» cos [0 — 4>}. 
 
 The discovery of this term is due to Professor Hansen ; 
 its coefBcient is extremely small, about 1"5"; but, being of 
 a totally different nature from those due to successive 
 approximations, it was thought desirable to examine it, and 
 the above investigation, which was communicated to me by 
 J. C. Adams, Esq.,* will be read with interest on account 
 of its elegance. 
 
 With respect to the foregomg investigation, perhaps the following remarks 
 
 dN 
 will not be superfluous : — The value of - - is not the actual velocity of N, 
 
 but its velocity relatively to the position of the node as determined when the 
 
 motion of the ecliptic is neglected ; its integral is therefore £N the change of 
 
 longitude due to this motion, and in this integration no constant is added, zero 
 
 dN 
 being taken for the mean value. The periodic forms of both —r- and SN shew 
 
 that they oscillate about mean values, the time of a complete oscillation being 
 • Kow Lowndean Professor of Astronomy in the University of Cambridge. 
 
MOTION OF THE ECLIPTIC. 101 
 
 that required by sin (JV-0) and cos {N—<p) to go through their cycle. This 
 relative motion of the node is analogous to that of a particle moving in a straight 
 line under the action of a force varying directly as the distance. 
 
 Similar remarks apply to the inclination. 
 
 It must also be borne in mind that the result obtained Sp is not the difference 
 between the latitude referred to the actual ecliptic and that referred to a fixed 
 plane, but is the difference between the calculated latitude referred to the actual 
 ecliptic on supposition of its being fixed, and the correct latitude referred to the 
 same actual ecliptic when its motion is taken into account. 
 
 We may obtain an approximate value of the coefficient 
 
 ceo by substituting for it — - , where a is the number of seconds 
 
 through which the ecliptic is deflected in one year = 0'48", 
 and n is the number of years in which the node of the moon's 
 
 orbit makes a complete revolution = 186; for then, — is 
 
 the angle described by the node in one year; therefore 
 
 «a . ^, .. - d {N- ^] . 
 
 —— is the ratio of lo : — ^-5 — - , supposing 9 to remain con- 
 
 27r at 
 
 stant, which is nearly the case ; therefore 
 na 9-3 x 0-48" 
 
 * This affords the solution of a problem proposed in the Senate-House 
 in the January Examioation of 1852. Question 21, Jan, 22. 
 
( 102 ) 
 
 CHAPTER VIII. 
 
 HISTORY OF THE LUNAR PROBLEM BEFORE NEWTON. 
 
 114. The idea which most probably suggested itself to 
 the minds of those men who first considered the motion of 
 the moon among the stars, was that it described a circle with 
 uniform velocity about the earth as a centre ; and this first 
 rough result is represented in our equations by neglecting 
 all small terms and writing 6 =j>t, u = a. 
 
 It must, however, have been very soon perceived that 
 the actual motion is far from being so simple, and that 
 the moon moves with very different velocities at different 
 times. 
 
 115. The earliest recorded attempts to take into account 
 the irregularites of the moon's motion were made by 
 Hipparchus (140 B.C.). He imagined the moon to move with 
 uniform velocity in a circle, of which the earth occupied, 
 not the centre, but a point nearer to one side. By a similar 
 hypothesis he had accounted for the irregularities in the 
 sun's motion, and his success in this led him to apply it also 
 to the moon. 
 
 It is clear that, on this supposition, the moon would seem 
 to move faster when nearest the earth or in perigee, and 
 slower when in apogee, than at any other points of her 
 orbit, and thus an apparent unequal motion would be 
 produced. 
 
EXCENTRIC AND EPICYCLE. 103 
 
 Let BAM be a circle, CA a radius, E a point \a AG 
 near C; GB, ED two parallel 
 lines making an angle a with 
 CA. 
 
 Suppose a body M to describe 
 this circle uniformly with an 
 angular velocity p, the time being 
 reckoned from the instant when 
 the body was at B, and the longi- 
 tude as seen from E being reckoned 
 from the line ED ; 
 therefore DEM =6, BCM =pt, 
 
 AEM = e-a, AGM=pt-a. 
 EG 
 Now Y^ is a small fraction, and if we represent it by e, 
 
 we shall have 
 
 sin M= -pjTTf sin AEM 
 CM 
 
 = e sin (0 — a) 
 
 = e sin [pt -i- M- a), 
 
 ,, esin (pt- a) 
 
 tan M= ^, ' , ; 
 
 1 — e cos [pt — a) 
 
 this would give M, and then d by the formula 6=pt + M, 
 
 This was called an excentric, and the value of e was called 
 
 the excentricity^ which, for the moon, Hipparchus fixed at 
 
 sma 1. 
 
 116. Another method of considering the motion was by 
 means of an epicycle, which led to the same result. 
 
 A small circle PM, with a radius equal to EC of previous 
 figure, has its centre in the circumference of the circle RPD 
 (which has the same radius as that of the excentric), and 
 moves round E with the uniform angular velocity p, the 
 body M being carried in the circumference of the smaller 
 
104 LUNAR THEORY. 
 
 circle, the radius PM remaining parallel to itself, or, which 
 
 is the same thing, revolving 
 
 from the radius PE with the 
 
 same angular velocity p^ so 
 
 that the angle EPM equals 
 
 PEA. 
 
 Now, when the angle AEP 
 equals the angle ACM of the 
 former figure, it is easily seen 
 that the two triangles EPM^ 
 ECM are equal, and there- 
 fore the distance EM and the angle AEM will be the same 
 in both, and the two motions will be identical. 
 
 117. The value of e being small, we find, rejecting e", &c., 
 J/=esin [pt — ai), 
 therefore ^ =^t + e sin {pt — a^. 
 
 If we reject terms of the second order in our expression 
 for the longitude, and make c = 1, we get. Art. (55), 
 
 6=pt + '2e sin [pt -a), 
 which will be identical with the above if we suppose the 
 excentricity of the excentric to be double that of the elliptic 
 orbit. 
 
 Ptolemy (a.d. 140) calculated the excentricity of the 
 moon's orbit, and found for it the same value as flipparchus, 
 viz. 
 
 sin 5° 1' = I'j, nearly. 
 
 The excentricity in the elliptic orbit is, we know, about ^*q, 
 and -jJj and -^-^ will pretty nearly reconcile the two values of 6 
 given above. This shews us, that for a few revolutions the 
 moon may be considered as moving iu an excentric, and her 
 positions in longitude calculated on this supposition will be 
 correct to the first order. 
 
EXCENTRIC AND EPICYCLE. 105 
 
 Her distances from the earth will not however agree ; for 
 the ratio of the calculated greatest and least distances would 
 
 be - — Jj^ or Jf , while that of the true ones would be It^ 
 
 or Y^. 
 
 It would, therefore, have required two different excentrics 
 to account for the changes in the moon's longitude and in 
 her radius vector. Changes in the latter could not, however, 
 be easily observed with the rude instruments the ancients 
 possessed, and it was very long before this inconsistency was 
 detected. 
 
 118. We have said that the moon's longitude, calculated 
 on the hypothesis of an excentric, will be pretty accurate for 
 a. few revolutions. 
 
 The data requisite for this calculation are, the mean 
 angular motion of the moon, the position of the apogee, and 
 the magnitude of the excentricity. 
 
 But it was known to Hipparchus and to the astronomers of 
 his time, that the point of the moon's orbit where she seems 
 to move slowest, is constantly changing its position among 
 the stars. Now this point is the apogee of Hipparchus's ex- 
 centric, and he found that he could very conveniently take 
 account of this further change by supposing the excentric 
 itself to have an angular motion about the earth in the same 
 direction as the moon, so as to make a complete revolution 
 in about nine years, or about 3° in each revolution.* 
 
 This motion of the apsldal line follows also from our 
 expression for the longitude, as shewn in Art. (75). It is 
 there, however, connected with an ellipse instead of an 
 
 * On the supposition of an epicycle, this motion of the apse could as easily 
 be represented by supposing the radius which connects the moon with the centre 
 of the epicycle to have this uniform angular velocity of about 3° in each revolu- 
 tion, and also in the same direction. 
 
 1' 
 
106 LUNAR THEORY. 
 
 excentric ; and though the discovery that the elliptic is the 
 true form of the fundamental orbit was not the next in the 
 order of time after those of Hipparchus, yet, as all the irre- 
 gularities which were discovered in the intervening seventeen 
 centuries are common both to Hipparchus's excentric and to 
 Kepler's ellipse, it will be as well for us to consider at once 
 this new form of the orbit. 
 
 Elliptic Form of the Orbit. 
 
 119. We need not dwell on the steps which led to this 
 great and important discovery. Kepler, finding that the 
 predicted places of the planet Mars, as given by the circular 
 theories then in use, did not always agree with the computed 
 ones, sought to reconcile these variances by other combina- 
 tions of circular orbits, and after a great number of attempts 
 and failures, and eight years of patient investigation, he 
 found it necessary to discard the excentrics and epicycles 
 altogether, and to adopt some new supposition. An ellipse 
 with the sun in the focus was at last his fortunate hypothesis? 
 which was found to give results in accordance with obser- 
 vation; and this form of the orbit was, with equal success, 
 afterwards extended to the moon : but the departures from 
 elliptic motion, due to the disturbing force of the sun, are, 
 in the case of the moon, much greater than the disturbances 
 of the planet Mars by the other planets. 
 
 In Keplers's hypothesis, then, the earth is to be considered 
 as occupying the focus of an ellipse, in the perimeter of which 
 the moon is moving, no longer with either uniform linear or 
 angular velocity, but in such a manner that the radius vector 
 sweeps over equal areas in equal times. 
 
 This agrees with our investigation of the motion of two 
 bodies. Art. (10). 
 
DISCOVERY OP EVECTION. 107 
 
 Evection. 
 
 120. The hypothesis of an excentric, whose apse line has 
 B progressive motion, as conceived by U.ipparcbus, served to 
 calculate with considerable accuracy the circumstances or 
 eclipses; and observations of eclipses, requiring no instru- 
 ments, were then the only ones which could be made with 
 sufficient exactness to test the truth or fallacy of the sup- 
 position. 
 
 Ptolemy (a.d. 140) having constructed an Instrument, by 
 means of which the positions of the moon could be observed 
 in other parts of her orbit, found that they sometimes agreed, 
 but were more frequently at variance with the calculated 
 places ; the greatest amount of error always taking place at 
 quadrature and vanishing altogether at syzygy. 
 
 What must, however, have been a source of great per- 
 plexity to Ptolemy, when he attempted to Investigate the 
 law of this new Irregularity, was to find that it did not 
 return In every quadrature, — In some quadratures it totally 
 disappeared, and in others amounted to 2° 39', which was 
 its maximum value. 
 
 By dint of careful comparison of observations, he found 
 that the value of this second Inequality in quadrature was 
 always proportional to that of the first in the same place, 
 and was additive or subtractlve according as the first was 
 so ; and thus, when the first Inequality In quadrature was at 
 its maximum or 6° 1', the second Increased it to 7° 40', which 
 was the case when the apse line happened to be In syzygy 
 at the same time.* 
 
 * It would Beem as if Hipparchus had felt the necessity for some further 
 modifioatiou of his first hypothesis, though he was unable to determine it ; for 
 there is an observation made by him on the moon in the position here specified 
 when the error of his tables would be greatest ; and at a time also when she 
 
108 LUNAR THEORY. 
 
 But if the apse line was in quadrature at the same time as 
 the moon, the second inequality vanished as well as the first. 
 
 The mean value of the two inequalities combined was 
 therefore fixed at 6° 20^'. 
 
 121. To represent this new inequality, which was sub- 
 sequently called the Evection^ Ptolemy imagined an excentric 
 in the circumference of which the centre of an epicycle moved 
 while the moon moved in the circumference of the epicycle. 
 
 The centre of the excentric and of the epicycle he supposed 
 in syzygy at the same time, and both on the same side of 
 the earth. 
 
 Thus, if E represent the earth, 
 
 S sun, 
 
 M moon, 
 
 c the centre of the excentric RKT in syzygy, 
 R, the centre of the epicycle, would also be in 
 
 syzygy- 
 
 Now conceive c, the centre of the excentric, to describe a 
 small circle about E m a, retrograde direction cc', while 5, 
 the centre of the epicycle, moves in the opposite direction, 
 
 was in the nonagesimal, bo that any error of longitude, arising from her yet 
 uncertain parallax, would be avoided. Ptolemy, who records the observation, 
 employs it to calculate the evection, and obtains a result agreeing with that 
 of his own observations. (See Delambre, Ast. Ancienne.) 
 
PTOLEMY'S HYPOTHESIS. 
 
 109 
 
 in such a manner that each of the angles 8'Ec\ S'EE may- 
 be equal to the synodical motion of the moon, that is, her 
 mean angular motion from the sun ; SES' being the motion 
 •of the sun in the same time. 
 
 Now we have seen, Art. (116), that the first inequality 
 was accounted for by supposing the epicycle RM to move 
 into the position rm, r and R being at the same distance 
 from E, and rm parallel to RM^ the first inequality being 
 the angle rEm. But when the centre of the epicycle is at 
 R\ and R'M' is parallel to r?w, the inequality becomes R'EM\ 
 and we have a second correction or inequality mEM'. 
 
 122. That this hypothesis will account for the phenomena 
 observed by Ptolemy, will be readily understood. 
 
 At syzygies, whether conjunction or opposition, the centres 
 of the excentric and epicycle are in one line with the earth 
 and on the same side of it ; the points r and R' coincide, as 
 also m and M'. Hence mEM' = 0. 
 
 At quadratures (figs, 1 and 2) c' and R' are in a straight 
 line on opposite sides of the earth, and therefore R' and r at 
 
 Fig. 1. Fig. 2. 
 
 their furthest distance. If, however, M' and m be at the 
 same time in this line, or, in other words, if the apse line 
 
 ♦ Tor simplicity we leave out of consideration the motion of the apse. 
 
110 LUNAR THEORY. 
 
 be in quadratures (fig. 1), the angle mEM' will still be zero, 
 or there will be no error in the longitude. But, if the apse 
 line is in syzjgj (fig. 2), the angle mEM' attains its greatest 
 value.* 
 
 Ptolemy, as we have said, found this greatest value to be 
 2° 39', the angle mEr being then 5° 1'. 
 
 123. Copernicus (a.d. 1543), having seen that Ptolemy's 
 hypothesis gave distances totally at variance with the obser- 
 vations on the changes of apparent diameter,t made another 
 and a simpler one which accounted equally well for the in- 
 equality in longtitude, and was at the same time more correct 
 in its representation of the distances. 
 
 Let E be the earth,^ OD an epicycle whose centre C de- 
 scribes the circle C" CC" about E with the moon's mean 
 angular velocity. 
 
 ■■-*' 
 
 Let G0,& radius of this epicycle, be parallel to the apse 
 
 * If Ptolemy had used the hypothesis of an excentric instead of an epicycle 
 for the first inequality of the moon, an epicycle would have represented the 
 second inequality more simply than his method did. Dr. Whewell's History of the 
 Inductive Sciences-^ vol. I., p. 230. 
 
 t See Delambre, Ast. Moderne, vol. I., p. 116. Whewell's History of Inductive 
 Sciences, vol. I., p. 395. 
 
COPEENICUa'S HYPOTHESIS, 111 
 
 line EA, and about as centre let a second small epicycle 
 be described, the radii CO and OMhe so taken that 
 
 CO-OM . ,. , .cq+OM . ,. ^, 
 
 -^ = sin5 l,and jj^ — =sin7 40. 
 
 The radius OM must now be made to revolve from the 
 radius OG twice as rapidly as EG moves from ES^ so that 
 the angle COM may be always double of the angle GE8. 
 
 From this construction, it follows that in syzygies the 
 angle GES being 0° or 180°, the angle COM is 0° or 360° ; 
 and therefore G and M are at their nearest distances, as in 
 the positions G ' and C" in the figure. Then GM= CO- OM, 
 and the angle GEM will range between 0° and 5° 1', the 
 greatest value being attained when the apse line is in quad- 
 rature. 
 
 When the moon is in quadrature GES =20° or 270°, 
 and, therefore, CGM= 180° or 540° and G and M are at 
 their greatest distance apart, as in the position C"; then, 
 GM= GO + OM, and the angle GEM will range between 
 0° and 7° 40', the former value when the apse line is itself 
 in quadrature, and the latter when it is in syzygy. 
 
 124. Thus the results attained by Ptolemy's construction 
 are, as far as the longitudes at syzygies and quadratures are 
 concerned, as well represented by that of Copernicus; and 
 the variations in the distances of the moon will be far more 
 exact, the least apparent diameter being 28' 45" and the 
 greatest 37' 33" ; whereas, Ptolemy's would make the greatest 
 diameter 1°.* 
 
 The values which modem observations give vary between 
 28' 48" and 33' 32". 
 
 • Delambre, Ast. Moderne. 
 
112 LUNAR THEORY. 
 
 125. It will not now be difficult to shew that the intro'- 
 duction of this small epicycle corresponds with that of the 
 term ^-^me sin {(2 - im -c)pt-'20 + a] in our value of ff. 
 
 For, referring to the preceding figure, we have 
 
 OEM= sin OEM = ^ sin OME 
 = ~^n[COM-AEM) 
 
 = ^8m[2.SEC-AEM) 
 UE 
 
 = -—- sin {2 (moon's mean long. — sun's long.) 
 UE 
 
 — (moon's true long. - long, of apse)j, 
 
 and OEM being a small angle whose maximum is 1° 19^', 
 we may write moon's mean longitude instead of the true 
 in the argument, and also EG for OE; therefore, 
 
 OEM= -^7~ sin {2 (moon's mean longitude — sun's longitude) 
 EL/ 
 
 — (moon's mean longitude — longitude of apse)} 
 
 = 79^' sin [2 {pt - {mpt + /3)} - {pt - [l-c)pt + a}] 
 
 = 4770" sin {(2 - 2m - c)pt - 2/3 + a}. 
 
 The value of the coefficient is from modem observations 
 found to be 4589-61". 
 
 126. In Art. (77), we have considered the eflfect of this 
 second inequality in another light, not simply as a small 
 quantity additional to the first or elliptic inequality, but as 
 forming a part of this first ; and therefore, modifying and 
 constantly altering the excentricity and the uniform pro- 
 gression of the apse line. 
 
 Boulliaud (a. D. 1645), by whom the term Evection was 
 first applied to the second inequality, seems to hint at some- 
 
HORROCK'S HYPOTHESIS. 113 
 
 thing of this kind in the rather obscure explanations of his 
 lunar hypothesis, which, never having been accepted, it would 
 be useless to give an account of it.* 
 
 In Ptolemy's theory, Art. (121), the evection was the 
 result of an apparent increase of the first lunar epicycle 
 caused by its approaching the earth at quadratures ; but, in 
 this second naethod, it is the result of an actual change in 
 the elements of the elliptic orbit. 
 
 D'Arzachel, an Arabian astronomer, who observed in 
 Spain about the year 1080, seems to have discovered the 
 unequal motion of the apsides, but his discovery must have 
 been lost sight of, for Horrocks, about 1640, re-discovered it 
 ' in consequence of his attentive observations of the lunar 
 
 * diameter : he found that when the distance of the sun from 
 ' the moon's apogee was about 45° or 225°, the apogee was 
 ' more advanced by 25° than when that distance was about 
 ' 135° or 315°. The apsides, therefore, of the moon's orbit 
 
 * were sometimes progressive and sometimes regressive, and 
 'required an equation of 12° 30', sometimes additive to their 
 
 * mean place and sometimes subtractive from it.'f 
 
 Horrocks also made the excentricity variable between the 
 limits -06686 and -04362. 
 
 The combination of these two suppositions was a means 
 of avoiding the introduction of Ptolemy's excentric or the 
 second epicycle of Copernicus : their joint effect constitutes 
 the evection. 
 
 • Apr6s aToir Stabli les mouTements et les gpoqnes de la lune, Boulliaud revient 
 i I'explication de I'eTection ou de la seconde megalith. Si sa thfeorie n'a pas fait 
 fortune, le nom du moins est reste. ' En meme temps que la lune avance aur son 
 ' cone autoor de la terre, tout le syst'eme de la lune eat deplace ; la terre emportant la 
 'lune, rejette loin d'elle I'apogee, et rapproche d'autant le perigee; mais cette 
 ' evection A, des homes fixees.' Delambre, Hist, de T Ast. Mod., torn. il. p. 167. 
 
 t Small's Astronomical Discoveries of Kepler, p. 307. 
 
 Q 
 
114 LUNAR THEORY. 
 
 Variation. 
 
 ] 27. After the discovery of the evection by Ptolemy, a 
 period of fourteen centuries elapsed before any further ad- 
 dition was made to our knowledge of the moon's motions. 
 Hipparchus's hypothesis was found sufficient for eclipses, and 
 ■when corrected by Ptolemy's discovery, the agreement be- 
 tween the calculated and observed places was found to 
 extend also to quadratures; any slight discrepancy being 
 attributed to errors of observation or to the imperfection 
 of instruments. 
 
 But when Tycho Brahe (a. d. 1580) with superior instru- 
 ments extended the range of his observations to all inter- 
 mediate points, he found that another inequality manifested 
 itself. Having computed the places of the moon for diflFer- 
 ent parts of her orbit and compared them with observation, 
 he perceived that she was always in advance of her com- 
 puted place from syzygy to quadrature, and behind it from 
 quadrature to syzygy ; the maximum of this variation taking 
 place in the octants, that is, in the points equally distant 
 from syzygy and quadrature. The moon's velocity therefore, 
 so far as this inequality was concerned, was greatest at new 
 and full moon, and least at the first and third quarters.* 
 
 * 'It appears that Jfohammed-Aboul-'Wefa-al-Bouzdjaiii, an Arabian astro- 
 nomer of the tenth century, who resided at Cairo, and observed at Bagdad in 975, 
 * discovered a third inequahty of the moon, in addition to the two expounded by 
 ' Ptolemy, the equation of the centre and the evection. This third inequality, the 
 'variation, is usually supposed to have been discovered by Tycho Brahe, six 
 
 'centures later In an almagest of Aboul-Wefa, a part of which exists in 
 
 ' the Royal Library at Paris, after describing the two inequalities of the moon, 
 'he has a Section IX., "Of the third anomaly of the moon called Muhazal or 
 
 ' Prosnemis" But this discovery of Aboul-Wefa appears to have excited 
 
 ' no notice among his contemporaries and followers ; at least it had been long 
 ' quite forgotten, when Tycho Brahe re-discovered the same lunai- inequality.' 
 Whcwcll's Hist, of Inductive Sciences, vol. 1. p. 243. 
 
DISCOVERY OF ANNUAL EQUATION. 115 
 
 Tycho fixed the maximum of this inequality at 40' 30". 
 The vahie which results from modern observations is 
 39' 30". 
 
 128. We have already two epicycles, or one epicycle and 
 an excentric, to explain the first two inequalities: by the 
 introduction of another epicycle or excentric, the variation 
 also might have been brought into the system ; but Tycho 
 adopted a different method:* like Ptolemy, he employed 
 an excentric for the evection, but for the first or elliptic 
 inequality be employed a couple of epicycles, and this 
 complicated combination, which it is needless further to 
 describe, represented the change of distance better than 
 Ptolemy's. 
 
 To introduce the variation, he imagined the centre of the 
 larger epicycle to librate backwards and forwards on the ex- 
 centric, to an extent of 40|-' on each side of its mean position ; 
 this mean place itself advancing uniformly along the excentric 
 with the moon's mean motion in anomaly ; and the libration 
 was so adjusted, that the moon was in her mean place at 
 syzygy and quadrature, and at her furthest distance from it 
 in the octants, the period of a complete libration being half 
 a synodical revolution. 
 
 Annual Equation. 
 
 129. Tycho Brah^ was also the discoverer of the fourth 
 inequality, called the annual equation. This was connected 
 with the anomalistic motion of the sun, and did not, like the 
 previous inequalities, depend on the position of the moon in 
 her orbit. 
 
 • For a fall description of Tycho's hypothesis, see Delambre, Hist, de V Ast. 
 Mod. torn. I. p. 1G2, and An Account of the Astronomical Discoveries of Kepler 
 by Robert Small, p. 130. 
 
116 LUNAR THEOEY. 
 
 Having calculated the position of the moon corresponding 
 to any given time, he found that the observed place was 
 behind her computed one while the sun moved from perigee 
 to apogee, and before it in the other half year. 
 
 Tycho did not state this distinctly, but he made a cor- 
 rection which, though wrong in quantity and applied in an 
 indirect manner, shewed that he had seen the necessity and 
 understood the law of this inequality. 
 
 He did not try to represent it by any new excentric or 
 epicycle, but he increased by (8m. 13s.) sin(suw'« anomaly) 
 the time which had served to calculate the moon's place;* 
 thus assuming that the true place, after that interval, would 
 agree with the calculated one. Now, as the moon moves 
 through 4' 30" in 8m. 13s., it is clear that adding (8m. 13s.) 
 sin [sun's anomaly] to the time is the same thing as sub- 
 tracting (4' 30") sin [suns anomaly) from the calculated lon- 
 gitude, which was therefore the correction virtually intro- 
 duced by Tycho-t Modern observations shew the coeflScient 
 to be 11' 9". 
 
 We have seen, Art. (82), how this inequality may be 
 inferred from our equations. 
 
 Reduction. 
 
 130. The next inequality in longitude which we have to 
 consider is not an inequality in the same sense as the fore- 
 going ; that is, it does not arise from any irregularity in the 
 motion of the moon herself in her orbit, but simply because 
 tliat orbit is not in the same plane as that in which the 
 
 * That is, the equation of time which he used for the moon differed by that 
 quantity from that used for the sun. 
 
 t Horrocks (ItiJ'J) made the correction in the same manner as Tycho, but so 
 increased it that the coiTesponding coefficient was 11' 51" instead of 4' 30". 
 Flamsteed was the first to apply the correction to the longtitude instead of the time. 
 
EEDUCTION. 117 
 
 longitudes are reckoned, so that even a regular motion in 
 the one would be necessarily irregular when referred to the 
 other. Thus if NMn be the moon's orbit and TiV»n the 
 ecliptic, and if M the moon be re- 
 ferred to the ecliptic by the great 
 circle Mm perpendicular to it, then 
 MN and mN are 0°, 90°, 180°, 270°, 
 and 360° simultaneously, but they 
 diflFer for all intermediate values : the diflference between 
 them ia called the reduction. 
 
 The difference between the longitude of the node and that 
 of the moon in her orbit being known, that is the side NM of 
 the right-angled spherical triangle NMm, and also the angle 
 N the inclination of the two orbits, the side Nm may be 
 calculated by the rules of spherical trigonometry, and the 
 difference between it and NM, applied with a proper sign 
 to the longitude in the orbit, gives the longitude in the 
 ecliptic. 
 
 Tycho was the first to make a table of the reduction 
 instead of calculating the spherical triangle. His formula 
 was 
 
 reduction = t&n'^I am2L — ^ tan*^Z sin4i, 
 
 where / is the inclination of the orbit and L the longitude of 
 the moon diminished by that of the node. 
 
 The first term corresponds with the term - ^k' sin2 {gpt — 7) 
 of the expression for 6. 
 
 Latitude of the Moon. 
 
 131. That the moon's orbit is inclined to the ecliptic was 
 known to the earliest astronomers, from the non-recurrence 
 of eclipses at every new and full moon; and it was also 
 known, siuces the eclipses did not always take place in the 
 same parts of the heavens, that the line of nodes represented 
 
118 LUNAR THEORY. 
 
 by Nn, in the preceding figure, Las a retrograde motion on 
 the ecliptic, iV" moving towards T. 
 
 Hipparchus fixed the inclination of the moon's orbit to 
 the ecliptic at 5°, which value he obtained by observing the 
 greatest distance at which she passes to the north or south 
 of some star known to be in or very near the ecliptic, as for 
 instance the bright star Regulus ; and by comparing the re- 
 corded eclipses from the times of the Chaldean astronomers 
 down to his own, he found that the line of nodes goes round 
 the ecliptic in a retrograde direction in about l8f years. 
 
 This result is indicated in our expression for the value 
 of the latitude by the term k sin (g9 — 7), as we have shewn 
 Art. (85). 
 
 132. Tycho Brah^ further discovered that the inclination 
 of the lunar orbit to the ecliptic was not a constant quantity 
 of 5° as Hipparchus had supposed, but that it bad a mean 
 value of 5° 8', and ranged through 9' 30" on each side of 
 this, the least inclination 4° 58|^' occurring when the node 
 was in quadrature, and the greatest 5° 17^' being attained 
 when the node was in syzygy.* 
 
 He also found that the retrograde motion of the node was 
 not uniform : the mean and the true positions agreed very 
 
 * Ebn Jounis, an Arabian astronomer (died a.d. 1008), whose works were 
 translated about 50 years since by Mons. Sedillot, states that the inclination of the 
 moon's orbit had been often observed by Aboul-Hassan-Aly-ben-Amajour about 
 the year 918, and that the results he had obtained were generally greater than 
 the b° of Hipparchus, but that they varied considerably^ 
 
 Ebn Jounis adds, however, that he himself had observed the inclination 
 several times and found it 5° 3', which leads us to infer that he always observed 
 in similar circumstances, for otherwise a variation of nearly 23' could scarcely 
 have escaped him. See Delambre, Hist, de V Ast. du Moyen Age, p. 139. 
 
 The mean value of the inclination is 5° 8' 55'4G",— the extreme values are 
 4° 67' 22" and 5° 20' 6". 
 
 The mean daily motion of the line of nodes is 3' 10'61", or one revolution in 
 6793-29 days, or 18 y. 218 d. 21 h. 22m. 46 a. 
 
CHANGE OF INCLINATION 119 
 
 well when they were ia syzygy or quadrature, but they 
 were 1° 46' apart in the octants. 
 
 By referring to Ai'<^- (87) we shall see that these correc- 
 tions, introduced by Tycho Brahe, correspond to the second 
 term of our expression for s. 
 
 Since Hipparchus could observe the moon with accuracy 
 only in the eclipses, at which time the node is in or near 
 syzygy, we see why he was unable to detect the want of 
 uniformity in the motion of the node. 
 
 133. To represent these changes in the position of the 
 moon's orbit, Tycho made the following hypothesis. 
 
 Let ENF be the ecliptic, K its pole, BA C a small circle, 
 
 having also K for pole and at a distance from it equal 
 to 5° 8'. Then, if we suppose A the pole of the moon's 
 orbit to move uniformly in the small circle and in the 
 direction BAG, the node N, which is at 90° from both A 
 and Kj will retrograde uniformly on the ecliptic, and the 
 inclination of the two orbits will be constant and equal to AK. 
 But instead of supposing the pole of the moon's orbit to 
 be at A, let a small circle abed be described with A as pole 
 and a radius of 9' 30" ; and suppose the pole of the moon's 
 orbit to describe this small circle with double the velocity 
 
120 LUNAR THEORY. 
 
 of the node in its synodical revolution which is accomplisbed 
 in about 346 days, in such a manner that when the node is 
 in quadrature the pole may be at a, the nearest point to K, 
 and at c the most distant point when the node comes to 
 syzygy, at d in the first and third octants, and at h in the 
 second and fourth, so as to describe the small circle in about 
 173 days, the centre A of the small circle retrograding mean- 
 while with its uniform motion. 
 
 By this method of representing the motion, we see that 
 
 1 1 • • 1 . ^the Inclina-,-. ,,„, „,, ,o,„i, 
 
 when node is m quadrature] . . , \Ka=b 8 —9* =4 58+ , 
 
 ^ ^ tion of the -^ ^ ,„„, „,, ,o.„T, 
 
 syzygy | ^^^^.^ .^ \Z-c=5°8'+9V=5°17^', 
 
 while at the octants it has its mean value 
 
 Again, with respect to the motion of the node, since N'\s 
 the pole of KaAc^ it follows that when In syzygy and quad- 
 rature, the node occupies its mean place ; in the first and 
 third octants, the pole being at d^ the node is before its 
 mean place by the angle dKA=l^' 30") cosec5° 8' = 1° 46', 
 nearly, and it is as much behind In the second and fourth 
 octants. 
 
 So that the whole motion of the node, and the correction 
 which Tycho had discovered, were properly represented by this 
 hypothesis, which is exactly similar to that which Copernicus 
 had imagined to explain the precession of the Equinox. 
 
{ 121 ) 
 
 SELECTION OF EXAMINATION QUESTIONS 
 
 PBOM COLLEGE AND SENATE-HOUSE PAPERS AND FROM THE 
 MODERATOBSHIP AND FELLOWSHIP EXAMINATIONS AT 
 TRINITY COLLEGE, DUBLIN. 
 
 1. Define the plane of the ecliptic and prove that, as 
 seen from the earth, sin X = ^j^^^ sinV nearly; \ being 
 the latitude of the sun and X' that of the moon. 
 
 2. Obtain the differential equation of the moon's radius 
 vector. 
 
 3. What is the principle on which the successive approxi- 
 mations to the moon's path are obtained ? 
 
 Compare the coefficients of. the principal term in the cen- 
 tral disturbing force with the principal term in the central force. 
 
 4. If in solving the equation 
 
 cPu _ P _T_du (cPu \ [ T 
 
 W^""' hw ~ h'u' dd "^ \M "^ "; i AV '*'' 
 
 we take as our first approximation u = a {l+e cos(0-a)}, 
 we obtain at our higher approximations terms containing 6 in 
 their coefficient. How is this defect avoided by taking 
 u = a {l+e cos(c0 — a)} as our first approximation? 
 
 Upon what principle do we approximate to the value of c ? 
 
 T 
 
 5. Shew that in the expansion of jj— 3 there will be a 
 
 hu 
 
 term - i^wiV sin { (2 - 2m - 2o) 5 - 2/3 + 2a} which would 
 rise to the second order in the longitude. 
 
 Shew further that the term with the same argument does 
 not appear actually in the longitude — assuming 
 M =a [I + e CQs[cd - a) +...+ ^me cos {(2 - 2m - c) - 2,8 + a} 
 +...+ »/me'cos{(2-2m-2c) e-2j8 + 2a}+...]. 
 
 K 
 
122 SELECTION OF EXAMINATION QUESTIONS. 
 
 6. The longitude of the moon contains a term of the form 
 
 A sin {(2 - 2m -c) pi -2/3 4 a} 
 where A is of the second order; find how it must have 
 arisen and determine A. 
 
 7. Investigate the following expression for the moon's 
 longitude as far as the second order, supposing the orbits 
 of the moon relative to the earth, and of the earth relative 
 to the sun, to be originally circles in the plane of the ecliptic, 
 
 e =pt + ^^m' sin {(2 - 2m) pt - 2/8}. 
 
 8. Suppose that whatever constants are involved in the 
 arguments 0, ^, i/r, &c. in the equation 
 
 L = A + Bs'md+Ca\n(li + Dsinylr + &c. 
 are perfectly known, and that observation gives the numerical 
 values of a considerable number of longitudes i,, L^, i,, &c., 
 corresponding to the known angles 0,,^^,'<ff^,&c., d^,(j>^,y{r , &c., 
 0^, <^„ i/f„ &c., &c., shew how the constants A, B, U, &c. 
 may be numerically determined in the two distinct cases 
 where the observations at our disposal are and are not 
 unlimited in number. 
 
 9. Explain how this process fails and may be modified 
 when two of the terms in question B sin 0, C sin <i> are nearly 
 synchronous in their periods. 
 
 10. Shew that the evection in longitude, viz. 
 
 i^yne sin {(2 - 2w - c) ;j« - 2/3 + a}, 
 may be represented as the joint eflfect of certain periodic 
 changes in the excentriclty of the lunar orbit and in the 
 mean longitude of its apse. 
 
 11. Assuming the usual notation of the Lunar Theory, 
 explain the physical meaning of the following equations : 
 
 s = k sin (^ — 7), 
 
 s = ^ sin {g0 — 7), 
 
 s = k BiD{g0-y) + ^mk sin{(2 -2»i-^) ^-2/3 + 7}. 
 
 12. Explain the efi'ect of the term m'a cos{(2-2m) 0—2/8)} 
 in the moon's radius vector; find the number of days in 
 
SELECTION OF EXAMINATION QUESTIONS. 123 
 
 the period of the resulting inequality and the ratio of the 
 axes of the oval orbit. 
 
 13. Investigate the effect of the annual equation on the 
 length of the lunar month. 
 
 Assuming »i=-075, e=-017, sin 13°30'=-23345, and the mean 
 period of a sidereal revolution of the moon = 27d. 7h. 43m. 128., 
 find the difference between a winter and a summer month. 
 
 14. Shew that the moon's orbit in space is always concave 
 towards the sun. 
 
 Assuming that the earth describes a circle about the sun, 
 and the moon a circle about the earth and in the same plane 
 with the earth's orbit, compare the curvatures of the moon's 
 orbit in space at her perihelion and aphelion. 
 
 15. Prove that if we go to the third order of approxi- 
 mation, the motion of the moon's apse in one revolution 
 of the moon equals f «" (1 — ^^to) 360.° 
 
 P T 
 
 16. The expressions for yf-^ and ^^-^ contain terms of 
 
 the form iA cos (6 — 6') and A sin [6 — ff) respectively ; com- 
 pute the terms resulting from them in the values of u and 6. 
 
 17. Copernicus represented the evection by an epicycle 
 superposed on the epicycle which represented the elliptic 
 inequality. From the construction given by him shew 
 geometrically that the evection vanishes when the moon'a 
 mean elongation from the sun is half her true anomaly, 
 and, in general, varies nearly as the sine of twice the 
 difference of these angles. 
 
 18. The secular change of Inclination of the actual ecliptic 
 to a fixed ecliptic being a" annually, shew that this will 
 give rise to an inequality in the moon's latitude whose type 
 
 is — jj— a"cos^, nearly; n being the number of years in 
 ztt 
 
 which the moon's nodes make a complete revolution, and 
 
 ^ the difference of longitude of the moon and of the ascending 
 
 node of the actual ecliptic. 
 
 W. METCALFE AND SON, PRINTBKS, CAMBRIDGE 
 
BY THE SAME AUTHOE. 
 
 I, 
 A TREATISE ON ASTRONOMY, for the use of 
 
 Colleges and Schools. 8to. cloth, 12s. 6d. 
 Macmillan and Co. 
 
 II. 
 
 A CHART AND DIAGRAM for Facilitating Great- 
 
 Circle Sailing, 3«. 6d. 
 
 With a Paper from Part ll., Vol. x. of the Cambridc/e 
 Philosophical Transactions. 
 
 III. 
 
 A TIME-AZIMUTH DIAGRAM for Determining the 
 
 Time-Azimuth without Calculation. 3s. 6d. 
 With a Pamphlet explaining its use and construction. 
 
 These Diagrams have been engraved by the Hydrographic Office, 
 Admiralty. Published by J. D. Potter, 31, Poultry, London, Agent for 
 the Admiralty Charts.