?

Therefore the orbit of G about S is very approximately an ellipse with S in the focus, and the plane of this ellipse is, as far as our investigations are concerned, a fixed plane when S is fixed.

This fixed plane is called the plane of the ecliptic, or simply the ecliptic.

16. A plane through the earth's centre parallel to the ecliptic will be the plane of reference we require (14) and will become a fixed plane when we bring the earth's centre to rest, the ecliptic then making small monthly oscillations from one side to the other of our fixed plane.

17. The sun will have a latitude always of the same name as that of the moon, and deducible from it, when

ES, EM, and the ratio of the masses of the earth and moon are known. For if S'EM' be this fixed plane through E, and S', G', M', the projections of S, G, M,

Now, from observation it is known that M is about go of E,

And as the moon's latitude never exceeds 5° 9′, the sun's latitude will always be less than 1".

Again, with respect to the sun's longitude: let Er be the direction of the first point of Aries,—that is, a fixed line in our plane of reference from which the longitudes of the bodies are reckoned. TES'' the sun's longitude.

The difference in the sun's longitude, as seen from E or from G, will be the angle ES'G'.

sin ES' G' =

EG'
ES'

sin @=

sin w 32400

approximately, if EG'S' = w;

therefore sin ES'G' never exceeds 3100

therefore ES'G' is a small angle not exceeding 7".

Also ES'- S'G' <EG' <3200 S'G'.

instead of S,

Now, by assuming the longitude and distance of the sun as seen from E to be the same as when seen from G, we commit the above small errors in the position of S; that is, we assume the sun to be at S" S'S" being drawn equal and parallel to G'E. If our object were the determination of the sun's position, it would be necessary to take this into account; but the consequent small errors introduced in the disturbance of the moon will clearly, on account of the great distance of the sun, be of a far higher order, and may therefore be neglected.

18. Hence we may assume that the motion of the sun about the earth at rest is an ellipse having the earth for its focus, and its equation

u' = a' {1 + e' cos (0' — 5)},

C

and we are safe that no appreciable error will ensue in the determination of the moon's place.*

* That is, as far as the three bodies alone are concerned ;-but, since the attractions of the planets may, and in fact do, disturb the elliptic orbit of the sun about G, the same cause will disturb the assumed orbit about E. A remarkable result of this disturbance is noticed in Appendix, Art. (99).

CHAPTER III.

RIGOROUS DIFFERENTIAL EQUATIONS OF THE MOON'S MOTION AND APPROXIMATE EXPRESSIONS OF THE FORCES.

E

19. The earth having been reduced to rest by the process described in Art. (9), its centre may be taken as origin of coordinates, the fixed plane of reference as plane of xy and the line ET as axis of x.

M

M'

Let r, be the coordinates of the projection M' of the moon on the plane xy, s the tangent of the moon's latitude MEM'. Also let the accelerating forces which act on the moon be resolved into these three:

P parallel to the projected radius M'E and towards the earth, T parallel to the plane xy, perpendicular to P and in the direction of increasing,

S perpendicular to the fixed plane and towards it.

1

20. These three equations for determining the moon's motion take the time t for independent variable, but it will be more convenient in the following process to consider the longitude as such, and our next step will be to change the independent variable from t to 0.