[1–2e cos (c0 − a) —− 2Σ„ − 2Σ, + fe’+ }e2 cos2 (c0− a) − −

+45me cos {(2- 2m - 2c) 0 −2B+2a}]

́1+ že2 + § k2 + m2 — 2e cos (c) — a)

Also, from Art. (45),

T

1-12 do = 1-4m2 cos {(2 — 2m) 0 — 28}

3

+me cos((2- 2m - 2c) 0-2ẞ+2α},

neglecting the other term of the third order, the coefficient of the argument not being small.

We have now to multiply these results together, and we see that the term having for argument (2-2m-2c)0-26+2a will disappear in the product. If we trace this term, we shall find that it arose in de, from retaining originally terms

T

3

1

of the fourth order, but in it arises from combining terms

hu2

not exceeding the third order. If, therefore, we had rejected terms beyond the third order indiscriminately, the expression

dt

do

1 would have contained this term, introduced by hu

and in t it would have been raised to the second order, and therefore formed an important part of its value instead of disappearing altogether from the expression. Hence the necessity for retaining such terms of the fourth order in

T

h'u3

3.

therefore pha" (1-e-k-m3) to the third order;

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

pt=0—2e sin (c0-a) + şe3 sin 2 (c0 − a)

no constant is added, the time being reckoned from the instant when the mean value of

explained in Art. (34).

vanishes, for the reasons

2.

51. The preceding equations U, 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

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) + 2e2 sin2 (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

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 — i̟m2 — e2+ e cos (cpt − a) + e2 cos2 (cpt—a)

- 4k2 cos2 (gpt — y)

+ m2 cos ((2-2m) pt-28}

+ me cos{(2-2m−c)pt—2B+a}}

...

..U;

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.

8

53. Similarly, the expression for a becomes s = k sin {(gpt− y) + 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 cos 2 (gpty)} to the second order,

= Ra

́1 − k2 — {m3 — e2 + e cos(cpt − a) +e* cos2 (cpt− a)

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

neglecting terms of the third order.

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.

If this be done, it is found that

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.

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 0 by the equation

2°

. 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' - 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

་

* En refléchissant sur les termes que doivent introduire toutes les 'quantités précédentes, on voit qu'il se peut glisser des cosinus de l'angle '0 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) avec des cosinus de m◊......

...... 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 dé'couvre 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.