50. We can now find the connexion between the longitude and the time to the second order,

1

haa {1 + e cos(c✪ − a) + Σ1⁄2 + Σ ̧}2 '

(Σ, being the sum of all the terms of the second, and Σ ̧

2e cos (c✪ − a) — 2Σ„ − 2Σ ̧ + že2 + že2 cos2 (c0 − a) + 45 me2 cos {{2-2m - 2c) 0 - 28 + 2a}]

́1 + že2 + 3 k2 + m2 — 2e cos (c0 − a)

2m2 cos {(2 — 2m) 0 – 2ẞ}

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 − 2ß + 2a will disappear in the product. If we trace this term, we shall T find that it arose in h2u3

de, from retaining originally terms of

1

the fourth order, but in

it arises from combining terms

hu

originally of the first and third orders. If, therefore, we had rejected terms beyond the third order indiscriminately, the ex

to the third order, 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

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

pt=0 - 2e sin (c0 − a) + şe2 sin2 (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).

2

51. The preceding equations U, S, O, 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 in the expressions for u and s.

= 2e {sin(cpt − a) + 2e sin (cpt — a) cos(cpt – a)} to the second order,

=

2e sin (cpt − a) + 2e2 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) + že2 sin 2 (cpt − a)

- 4k2 sin 2 (gpt - y)

+ m2 sin {(2 − 2m) pt — 2ẞ}

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

+ 15me cos {(2−2m−c) pt−2B+a}

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

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

at the time t,

radius of earth

R

Ru (1 − s) to the third order,

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

but P

k2 – ¿m2 – e2 + e cos(cpt − a) + e2 cos 2 (cpt — a)

+ m2 cos {(2-2m) pt - 2ẞ}

= the portion which is independent of periodical terms, = Ra (1-k - {m2 — e2);

of

55. Here we terminate our approximations to the values u, 8, 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 values of P, T, and S, given in Art. (23), would no longer be sufficiently accurate, but we should have to recur to more 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 0 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.