CHAPTER V.

NUMERICAL VALUES OF THE COEFFICIENTS.

59. 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.

60. First method. By particular observations of the sun and moon (i.e. by observations made when they occupy particular and selected positions), and also by observations separated by very long intervals, such, for instance, as ancient and modern eclipses, the values of the constants p, m, a, B, y, 5, which enter into the arguments, and of the additional ones which enter into the coefficients of the terms in the previous developments, may be obtained with great accuracy, and by their means, the coefficients themselves; c and g being also known in terms of the other constants. These may properly be called the theoretical values of the coefficients, the only recourse to observation being for the determination of the numerical values of the elements.

61. 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 ob

tained, 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 being thus obtained, they are then, by the method of least squares or some analogous process, reduced to as many as there are coefficients 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. (58), each of the numerous equations would consist of 130 terms, and these would have to be reduced to 129 equations of 130 terms each.

62. Third method. When the constants which enter into the arguments have been determined by the first method, we may obtain any one of the coefficients independently of all the others by the following process, provided the number of observations be very great.

Let the form of the function be

V=A+B sin 0 + C sino + &c.,

and let it be required to determine the constants A, B, C, &c. separately; 0, 4, &c. being functions of the time.

Let the results of a great number of observations corresponding to values 01, 02, 0, &c., 41, 42, 43, &c., be V1, V2, V1 &c.; so that

19

29

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 there-
fore, that if we add all the equations together these terms
will disappear;

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 ber terms in the first set
and s terms in the second, and let V', V", ...... V' be the
values of V corresponding to positive values of sin, which
values we may assume to be uniformly distributed from
sino to sin, and therefore to be sin de, sin 280,
where r.80 =T.

...... 1119

sinr.80,

And, again, let V, V, V... V, be the values of V corresponding to the negative values of sine, viz.; siņ (— ▲0), sin(-s.A), where s. 40.

sin (-240),

......

V' = A+B sin 80

+C sind' +..., V, A-B sin A0

=

Then,

+C sino, ..., V" A+B sin 2.80+ C sin p" +..., VA-B sin 2.A0+ C sino, + ..., =

įr′ = A + B sinr.80 + C′ sind +...; V, 4-B sins. A + C sind, + ...,

V

therefore

= A

therefore

V'+V"+...+V'=r.A+ BΣ" (sin 0) V+V+...+V=S.A-BΣ," (sin0)

and in a similar manner may each of the coefficients be independently determined.*

For further remarks on this method, see Appendix, Art. (104).

* If r and s are not sufficiently great to allow us to substitute f" sinode for " sine.de, we must proceed as follows:

п

V'+V" + · + V* = rA + B (sin d0+ sin 280 + + sin rò0)

......

CHAPTER VI.

PHYSICAL INTERPRETATION.

63. 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 coefficients in these equations may be determined from observation; and we have, moreover, 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, and also whether 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.

* 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 introductory treatise. For information on these points reference may be made to Airy's Figure of the Earth. Pontécoulant's Système du Monde, vol. IV.