Ρ

therefore pha2 (1 — §e* — §k2 — m2) to the third order;

therefore, multiplying by p and integrating, we get, still to the second order,

pt=0-2e sin (c0− a) + že2 sin 2 (c0 − a)

no constant is added, the time being reckoned from the instant when the mean value of vanishes, for the reasons explained in Art. (34).

29 2

51. The preceding equations U2, S2, give the reciprocal 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 0 in terms of t.

To do this, we must reverse the series pt=0&c., and then substitute the value of 0 in the expressions for u and s. 0=pt+2e sin(c0 - a) to the first order

Now

=pt+2e sin (cpt-a)... ;

therefore co-a=cpt—a+2e sin (cpt- a) to the first order, 2e sin (c0 − a) = 2e {sin (cpt − a) + 2e sin(cpt - a) cos(cpt — a)} to the second order,

=2e sin (cpt − a) + 2e3 sin 2 (cpt — a).................................. ;

and as 0 and pt differ by a quantity of the first order, they may be used indiscriminately in terms of the second order; therefore

0=pt+2e sin(cpt-a)+ şe2 sin2 (cpt- a)

52. In the value of u given in Art. (48), substitute pt for in terms of the second order, and pt +2e sin (cpt - a) in the term of the first order; then

u=a

· 1—3k2 —¿m2 — e2+e cos (cpt− a) +e2 cos 2 (cpt—a)

the other terms in the value of u in Art. (48), which were there retained only for the sake of subsequently finding t, being of the third order, are here omitted.

53. Similarly, the expression for s becomes

s=k sin {(gpty) + 2e sin (cpt-a)}

The expression for s is more complex in this form than when given in terms of the true longitude 0.

54. If P be the moon's mean parallax, and II the parallax

= Ru {1 − 4k2 + 4k2 cos2 (gpty)} to the second order,

́1 − k2 — ¿m2 — e2 + e cos (cpt− a)+e2 cos2 (cpt—a)

but P= the portion which is independent of periodical terms,

55. Here we terminate our approximations to the values of u, s, and 0. 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. (29), and the approximate values of P, T, and S, given in Art. (23), would no longer be sufficiently accurate, but we should have to recur to the exact values, and from them obtain terms of an order beyond those already employed.

These terms of the fourth order become of the third order in the value of u, and therefore also of t, the coefficient of being near unity.

We shall see further on (Appendix, Art. 97), to what purpose a knowledge of the existence of these terms has been applied.

56. The process followed in the preceding pages is a sufficient clue to what must be done for a higher approximation.

2°

The coordinates u' and ' of the sun's position are, by the theory of elliptic motion, known in terms of the time t, and t is given in terms of the longitude by the equation →. Hence u' and ' can be obtained in terms of ; but it will be necessary to take into account the slow progressive motion of the sun's perigee, which we have hitherto neglected. This will be done by writing c'e' - 5 for 0' — 5, c' being a quantity which differs very little from unity.*

These values of u', ', together with those of u and s in terms of e, as given by U, and S, are then to be substituted in the corrected values of the forces, and thence in the

2

......

* En refléchissant sur les termes que doivent introduire toutes les ' quantités précédentes, on voit qu'il se peut glisser des cosinus de l'angle ⚫ dont nous avons vu le dangereux effet d'amener dans la valeur de u des arcs au lieu de leurs cosinus; de tels termes viendront, par exemple, de 'la combinaison des cosinus de (1-m) 0 avec des cosinus de me.. ...... Pour éviter cet inconvenient qui ôterait à la solution précédente 'l'avantage de convenir à un aussi grand nombre de révolutions qu'on vou'drait, et la priverait de la simplicité et de l'universalité si précieuses en ' mathématiques, il faut commencer par en chercher la cause. Or, on découvre facilement que ces termes ne viennent que de ce qu'on a supposé 'fixe l'apogée du soleil, ce qui n'est pas permis en toute rigueur, puisque ' quelque petite que soit sur cet astre l'action de la lune, elle n'en est pas 'moins réelle et doit lui produire un mouvement d'apogée quoique très 'lent à la vérité.' Clairaut, Theorie de la Lune, p. 55, 2me Edition.

differential equations. The integrations being performed as before will give the values of u, s, and t in terms of to the third order, and from these, as in Arts. (51), (52), and (53), may be obtained and in terms of t.

57. More approximate values of c and g are obtained at the same time, by means of the coefficients of cos (c✪ − a) and sin (gy) in the differential equations, (see Appendix, Arts. 94 and 95).

58. The values to the fourth order are then obtained from those of the third by continuing the same process, and so 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 in terms of t contains 128 periodical terms, without including those due to the disturbances produced by the planets. The coeffia' cients of these periodical terms are functions of m, e, e', é', a,

a

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 Pontécoulant, Système du Monde, tom. IV. p. 572.