Now, from observation it is known that M is about 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'0' the sun's longitude.

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

therefore sin ES' G' never exceeds 300,

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

Also ES'- S'G' <EG' <32100 S'G'.

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" instead of 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')},

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.

ག

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, O 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.

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.

This is called the differential equation of the moon's radius

- Hu2 u

2

d's

= Hu

dt

d'u

dH

+H

u2

de ᏧᎾ

и

therefore Ps-S= Hus + +H

de

dei

dH ds

de

из

de

dei

This is the differential equation of the moon's latitude.

If the three equations (a), (B), (7) could be integrated under these general forms, then, since they are perfectly rigorous, the problem of the moon's motion would be completely solved; for as only four variables u, 0, s, and t are involved (the accelerating forces P, T, and S are functions of these), the values of three of them, as u, 0, s, could be