But Hipparchus had found, and all modern observations confirm his result, that the motion of the apse is about 3° in each revolution of the moon. See Art. (112).

This difference arises from our value of c not being represented with sufficient accuracy by 1-4m2.

Newton himself was aware of this apparent discrepancy between his theory and observation, and we are led, by his own expressions (Scholium to Prop. 35, lib. III. in the first edition of the Principia), to conclude that he had got over the difficulty. This is rendered highly probable when we consider that he had solved a somewhat similar problem in the case of the node; but he has nowhere given a statement of his method: and Clairaut, to whom we are indebted for the solution, was on the point of publishing a new hypothesis of the laws of attraction, in order to account for it, when it occurred to him to carry the approximations to the third order, and he found the next term in the value of c nearly as considerable as the one already obtained. See Appendix, Art. (94). c = 1-3m2 - 225 m3 ;

.. 1 − c = {m2 + 325m3 = 3m2 (1 + 75m) ;

... (1 − c) 360° = (1+74) (value found previously)

thus reconciling theory and observation, and removing what had proved a great stumbling-block in the way of all astronomers.*

When the value of c is carried to higher orders of approximation, the most perfect agreement is obtained.

The motion of the apse line is considered by Newton in his Principia, lib. I., Prop. 66, Cor. 7.

We shall consider this term in two different ways.
Firstly, by itself, as forming a correction on pt.

0 = pt + 15me sin {(2 - 2m — c) pt − 2B+ a}.

Let >=

α

then

=

pt mpt + B

= moon's mean longitude at time t, = sun's

= (1 − c) pt + a = mean longitude of apse

=

0 = pt + 15me sin [2{ pt — (mpt + B) } — { pt − (1 − c) pt+a}]

The effect of this term will therefore be as follows:

In syzygies

0 = pt me sin( − a');

or the true place of the moon will be before or behind the mean, according as the moon, at the same time, is between apogee and perigee or between perigee and apogee.

In quadratures = pt + me sin (-a),

and the circumstances will be exactly reversed.

In both cases, the correction will vanish when the apse happens to be in syzygy or quadrature at the same time as the moon.

In intermediate positions, the nature of the correction is more complex, but it will always vanish when the sun's longitude is equal to the arithmetical mean between those of the moon and apse, or when it differs from it by any multiple D + a - r.90°, 2

of 90°; for if © =

sin [2 (D − ©) – (D − a')] = sin ( » + a' −20)

70. The other and more usual method of considering the effect of this term is in combination with the two terms of the elliptic inequality, as follows:

To determine the change in the position of the apse and in the eccentricity of the moon's orbit produced by the evection.

Taking the elliptic inequality and the evection together, we have

Let a' be the longitude of the apse at time t on supposition of uniform progression,

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

+ 15 me sin{cpt — a + 2 (a' — ©)} ;

and the second and fourth terms may be combined into one,

E2 = e2 {1 + 1m cos 2 (a' — ©)}2 + e2{ 15m sin 2 (a' — ©)}2;

or, approximately,

=

15m sin 2 (a' — ©),

E=e{1+m cos 2 (a' - )}.

The term e sin 2 (cpt — a) will, therefore, to the second order, be expressed by

E' sin 2 (cpt-a + d),

and the longitude becomes

0 = pt +2E sin (cpt − a +8) + §E2 sin 2 (cpt — a + d),

or 0 = pt + 2E sin( pt − a′ +8) + §E2 sin 2 ( pt − a' + 8) ; but the last two terms constitute elliptic inequality in an orbit whose eccentricity is E and longitude of the apse a-8; therefore the evection, taken in conjunction with elliptic inequality, has the effect of rendering the eccentricity of the moon's orbit

variable, increasing it by me when the apse-line is in syzygy, and diminishing it by the same quantity when the apse-line is in quadrature; the general expression for the increment being

15 me cos 2 (a' — O).

=

And another effect of this term is, to diminish the longitude of the apse, calculated on the supposition of its uniform progression, by the quantity 8 m sin 2 (a); so that the apse is behind its mean place from syzygy to quadrature, and before it from quadrature to syzygy.

*

The cycle of these changes will evidently be completed in the period of half a revolution of the sun with respect to the apse, or in about of a year.

71. The period of the evection itself, considered independently of its effect on the orbit, is the time in which the - c) pt — 2B + a will increase by 2π. argument (2 - 2m

Newton has considered the evection, so far as it arises from the central disturbing force, in Prop. 66, Cor. 9 of the Principia.

Variation.

72. To explain the physical meaning of the term

m2 sin {(2 – 2m) pt − 2ß},

in the expression for the moon's longitude,

0 = pt + m2 sin {(2 – 2m) pt — 2ß}.

* The change of eccentricity and the variation in the motion of the apse follow the same law as the abscissa and ordinate of an ellipse referred to its centre: for if Eex and dy, then

Let represent the moon's mean longitude at time t,

which shews that from syzygy to quadrature, the moon's true place is before the mean, and behind it from quadrature to syzygy; the maximum difference being m" in the octants. The angular velocity of the moon, so far as this term is concerned, is

= p {1+ 4m2 cos 2 (D)}, nearly,

which exceeds the mean angular velocity p at syzygies, is equal to it in the octants, and less in the quadratures.

This inequality has been called the Variation, its period is the time in which the argument (2 – 2m) pt – 2ẞ will increase by 2π ;

mean synodical month

142 days, nearly.*

2

73. The quantity m2 is only the first term of an endless series which constitutes the coefficient of the variation, the other terms being obtained by carrying the approximation to a higher order. It is then found that the next term in the coefficient is m3, which is about of the first term; and as there are several other important terms, it is only by carrying the approximation to a high order (the 5th at least) that the value of this coefficient can be obtained with sufficient accuracy from theory. In fact, m2 would give a coefficient of 28′ 32" only; whereas the accurate value is found to be 39′ 30′′.

The same remark applies also to the coefficients of all the other terms.

• The accurate value is 14.765294 days.