RADIUS VECTOR.

83. To explain the physical meaning of the terms in the value of u.

We shall, for the explanation, make use of the formula which gives the value of u in terms of the true longitude, Art. (48).

Firstly, neglecting the periodical terms, we have for the mean value

The term - m2, which is a consequence of the disturbing effect of the sun, shews that the mean value of the moon's radius vector, and therefore the orbit itself, is larger than if there were no disturbance.

Elliptic Inequality.

84. To explain the effect of the term of the first order,

This is the elliptic inequality, and indicates motion in an ellipse whose eccentricity is e and longitude of the apse a+(1−c) 0; and the same conclusion is drawn with respect to the motion of the apse as in Art. (66).

Evection.

85. To explain the physical meaning of the term

15mea cos {(2 - 2m − c) 0 − 2B + a}.

This, as in the case of the corresponding term in the longitude, is best considered in connexion with the elliptic inequality, and exactly the same results will follow.

Thus calling ), O, and a' the true longitudes of the moon, sun, and apse, the latter calculated on supposition of uniform motion, these two terms may be written,

u = a[1 + e cos()− a') + 15 me cos { ) − a' + 2 (a' — ©)}]

E cosde+me cos2 (a),

These are identical with the equations of Art. (70).

Variation.

86. To explain the effect of the term m3a cos {(2− 2m) 0—2B}, u = a[1+m2 cos {(2 – 2m) 0 – 2ẞ}]

As far as this term is concerned, the moon's orbit would be an oval having its longest diameter in quadratures and least in syzygies. Principia, lib. I., prop. 66, cor. 4. The ratio of the axes of the oval orbit will be

87. The last important periodical term in the value of u is

This, when increased by a constant, is approximately the difference between the values of u in the orbit and in the ecliptic.

For if u, be the reciprocal of the value of the radius

88. The remaining terms in the value of u are of the third order, and therefore very small: one of these corresponds to the annual equation in longitude Art. (75), where it is of the second order, having increased in the course of integration.

Periodic time of the Moon.

89. We have seen, Art. (65), that the periodic time of the moon is greater than if there were no disturbing force; but this refers to the mean periodic time estimated on an interval of a great number of years, so that the circular functions in the expression are then extremely small compared with the quantity pt which has uniformly increased.

When, however, we consider only a few revolutions, these terms may not all be neglected. The elliptic inequality and the evection go through their values in about a month, the variation and reduction in about half-a-month; their effects, therefore, on the length of the period can scarcely be considered, as they will increase one portion and then decrease another of the same month.

But the annual equation takes one year to go through its cycle, and, during this time, the moon has described thirteen revolutions; hence, fluctuations may, and, as we shall now

shew, do take place in the lengths of the sidereal months during the year.

We have, considering only the annual equation, Art. (75),

pt=0+3me' sin (m0+B-).

Let T be the length of the period, then when ◊ is increased by 2, t becomes t+T;

therefore p (t+T) = 2π+0+3me' sin (2mπ+m0+B−5),

whence

therefore

pT=2π+6me' sinmπ cos(mπ+mo+B-C);

6me'

T= mean period + sinmπ cos(0–),

Ρ

where me+B+mπ sun's longitude at the beginning

=

of the month + mπ

= sun's longitude at the middle of the month.

Hence T will be longest when

-=0,

and shortest when O-=;

or T will be longest when the sun at the middle of the month is in perigee, and shortest when in apogee; but, at present, the sun is in perigee about the 1st of January, and apogee about the 1st of July; therefore, owing to annual equation, the winter months will be longer than the summer months, the difference between a sidereal month in January and July, from this cause, being about 20 minutes.

90. All the inequalities or equations, which our expressions contain, have thus received a physical interpretation. They were the only ones known before Newton had established his theory, but the necessity for such corrections was fully recognized, and the values of the coefficients had already been pretty accurately determined; still, with the exception of the reduction, which is geometrically necessary, they were corrections empirically made, and it was scarcely to be ex

pected that any but the larger inequalities, viz. those of the first and second orders which we have here discussed, could be detected by observation: we find, however, that three others have, since Newton's time, been indicated by observation before theory had explained their cause. These arethe secular acceleration, discovered by Halley; an inequality, found by Mayer, in the longitude of the moon, and of which the longitude of the ascending node is the argument; and finally an inequality discovered by Bürg, which has only of late years obtained a solution. For a further account of these, as also of some other inequalities which theory has made known, see Appendix, Arts. (99), (100), (101), (102).