Reduction.

82. We may now consider the term which we had neglected (Art. 76) in the expression for the longitude, namely,

Let N be the position of the node when the moon's longitude is 0, M the place of the

moon, m the place referred to the ecliptic.

Therefore m = 0,

N=y— (g− 1) 0,

Nm = 90 −y,

tan N=k.

The right-angled spherical triangle NMm gives

M

m

or, since both N and NM-Nm are small,

N

=

2

therefore NM-Nm=4k2 sin 2(g0—y) = 4k2 sin 2 (gpt—y), nearly.

Hence this term, which is called the reduction, is approximately the difference between the longitude in the orbit and the longitude in the ecliptic.

therefore, the accelerating forces on the moon al

+ (MG+GE) ......... in direction

(GM' + GE') cos(0-0').........parallel to

M'E' cos

r

{1+ cos 2 (0-0')}

{+ cos 2 (0-0')},

24. The differential equations in Art. (20), when t values of the forces are substituted in them, would cont a new variable e', but we shall find means to establis connexion between t, 0, and e', which will enable us to minate it.

They will, however, be still incapable of solution exc

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

u = a (1 − 3k2 — {m2).

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 — 2ß + 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 ), ©, 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}]

= a[1+m3 cos2() - ©)].

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

1+ m2

1-m2

= 38 nearly, m being '0748.

=

See Principia, lib. III., prop. 28.

Reduction.

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.