NUMERICAL VALUES.

53

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 " sinodo for 2 sin 0.00, we must proceed as follows:

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.

It is true that our expressions, composed of periodic terms, are nothing more than translations into analytical language of the epicycles of the ancients;† but they are evolved directly

* 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. † See Whewell's "History of the Inductive Sciences."

DISCUSSION OF THE LONGITUDE.

55

from the fundamental laws of force and motion, and as many new terms as we please may be obtained by carrying on the same process; whereas the epicycles of Hipparchus 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, with the means of observation they possessed.

LONGITUDE OF THE MOON.

64. We shall firstly discuss the expression for the moon's longitude, as found Art. (51).

0 = pt + 2e sin (cpt − a) + že2 sin 2 (cpt — a)

The mean value of 0 is pt; and in order to judge of the effect

of

any of the small terms, we may consider them 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 combinations in explaining the elliptic inequality and the evection, Arts. (66) and (70); but in the remaining inequalities each term of the expression will form a correction to be considered by itself.

65. Neglecting all the periodical terms, we have

0 = pt,

de

dt

=

P1

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 period of a revolution

2π

being which is, therefore, the expression for a mean sidereal " Ρ

month, or about 271 days.*

The value of p is, according to Art. (50), given by

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.

66. We shall next consider the effect of the first three terms together: the effect of the second alone, as a correction of pt, will be discussed in the Historical Chapter, Art. (109).

0 = pt + 2e sin (cpt- a) + že sin2 (cpt - a),

which may be written

0=pt + 2e sin [pt−{a+ (1−c) pt}] + že2 sin2 [pt - {a+(1−c) pt}]. But the connexion between the longitude 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 :

Ꮎ = nt + 2e sin (nt — a') + že2 sin 2 (nt — a'),

* The accurate value was 27d. 7h. 43m. 11.261s. in the year 1801. See Art. (99).

MOTION OF THE APSE.

57

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 has a progressive motion in longitude, uniform, and equal to (1 − c) p.

67. The two terms 2e sin (cpt- a) + že sin 2 (cpt-a) constitute the elliptic inequality, and their effect may be further illustrated by means of a diagram.

Let the full line AMB (fig. 8) represent 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 Er from which the longitudes are reckoned.

The angle YEA, the longitude of the apse, is then a. At the time t, when the moon's mean longitude is TEM = pt, the apse line will have moved in the same direction through the angle AEA' = (1- c) TEM, and the orbit will have taken the position indicated by the dotted ellipse; and the true place of the moon in this orbit, so far as these two terms are concerned, where

will be m,

= 2e sin A'EM(1+ že cos A'EM);

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.

68. The angular velocity of the apse is (1-c) p, or, if for c we put the value found in Art. (48), the velocity will be 3m2p. Hence, while the moon describes 360°, the apse should describe m2.360° = 13° nearly, m being about.

* The epoch & which appears in the expression of Art. (13) is here omitted; a proper assumption for the origin of t, as explained in Art. (34), enabling us to avoid the ɛ.