time as the true one, both passing through the apse at the same instant, then nt+-a is called the mean anomaly, ɛ being a constant depending on the instant when the body is at the apse, its value being also equal to the angle between the prime radius and the uniformly revolving one when t=0.

Thus, if MY be the fixed line or prime radius,

therefore, mean anomaly = AMμnt + ε — α,

true anomaly = AMM' =TMMM'-MA=0-a.

12. To express the mean anomaly in terms of the true in a series ascending according to the powers of e, as far as e2.

* The introduction of the epoch is avoided in the Lunar theory by a particular assumption (Art. 34); but in the Planetary it forms one of the elements of the orbit.

(1 — §e2) {1 — 2e cos(0 − a) + 3e2 cos2 (0 − a)}

= {(1 − 2e cos (0 − a) + §e cos2 (0 − a)} ;

n

therefore nt += 0-2e sin (0-a) + e2 sin2 (0-a),

or (nt+e-a)=(0 − a) — 2e sin (0 − a) + že2 sin 2 (0 − a),

the required relation.

13. To express the true anomaly in terms of the mean to the same order of approximation.

0 — a=nt + e − a + 2e sin (0 — a) — že2 sin 2 (0 — a) ...........(1);

.. 0―a=nt + ε- a first approximation.

Substituting this in the first small terms of (1), we get

0-a=nt +ε―a + 2e sin (nt+e—a)... a second approximation.

Substitute the second approximation in that small term of (1) which is multiplied by e, and the first approximation in that multiplied by e; the result will be correct to that term, and gives

0−a=nt +ε—a + 2e sin {nt + e − a + 2e sin (nt + e − a)}

+4e3 cos(nt + ɛ − a) sin(nt + e − a) — şe2 sin 2 (nt+ɛ—α)

=nt+e―a+2e sin (nt + e − a) + že2 sin 2 (nt + e − a),

the required relation.

The development could be carried on by the same process to any power of e, the coefficient of e3 would be found

13 sin 3 (nt + e − a) — — sin (nt + ε — α)),

but in what follows we shall not require anything beyond e2.

Problem of Three Bodies.

14. In order to fix the position of the moon with respect to the centre of the earth, which, by means of the process described in Art. (9), is reduced to and kept at rest, we must have some determinate invariable plane passing through the earth's centre to which the motion may be referred.

The plane which passes through the earth's centre and the direction of the sun's motion at any instant is called the true ecliptic; and, as a first rough result of calculation, obtained on supposition of the sun and earth being the only bodies in the universe, this plane, in which, according to the last section, an elliptic orbit would be described, is a fixed plane: but this is no longer the case when we take into account the disturbances produced by the moon and planets, and it becomes necessary to substitute some other plane of reference unaffected by these disturbances.

Theory teaches us that such a plane exists,* but as its

* See Poinsot, "Théorie et détermination, de l'équateur du système solaire,” where he proves that an invariable plane exists for the solar system, that is, a plane whose position relatively to the fixed stars will always be the same whatever changes the orbits of the planets may experience; but as its position depends on the moments of inertia of the sun, planets, and satellites, and therefore on their internal conformation, it cannot be determined à priori, and ages must elapse before observation can furnish sufficient data for doing so à posteriori.

This result Poinsot obtains on the supposition that the solar system is a free system; but it is possible, as he furthermore remarks, nay probable, that the stars exert some action upon it, it follows that this invariable plane may itself be variable, though the change must, according to our ideas of time and space, be indefinitely slow and small.

determination can only be the work of time, the following theorem will supply us with a plane whose motion is extremely slow, and it may for a long period and to a degree of approximation far beyond that to which we shall carry our investigations, be considered as fixed and coinciding with its position at present.

15. The centre of gravity of the earth and moon describes relatively to the sun an orbit very nearly in one plane and elliptic; the square of the ratio of the distances of the moon and sun from the earth being neglected.*

S

G

w

E

M

Let S, E, M be the centres of the sun, earth, and moon, G the centre of gravity of the last two. Now the motion. of G is the same as if the whole mass E+ M were collected there and acted on by forces equal and parallel to the moving forces which act on E and M. The whole force on G is therefore in the plane SEM; join SG.

Let 2 SGM=w, and let m' be the sun's absolute force.

This ratio is about and, as we shall see Art. (21), such a quantity we shall consider as of the 2nd order of small quantities, and its square therefore of the 4th order. Our investigations are carried to the 2nd

order only.

GM GE

=-3 (accelerating force of sun on G) SG SG

=0 according to the standard of approximation adopted.

Hence the only force on G is a central force tending to S, and therefore the motion of G will be in one plane.

* In strictness it would be necessary, since we have brought S to rest, to apply to both M and E, and therefore to G, accelerating forces equal and opposite to those which E and M themselves exert on S; but the mass of S is so large compared with those of E and M, that we may safely neglect these forces in this approximate determination of the path of G, the error being of a still higher order than that introduced