LUNAR THEORY.

CHAPTER I.

INTRODUCTION.

BEFORE proceeding to the consideration of the moon's motion, it will be desirable to say a few words on the law of attractions, and on the peculiar circumstances which enable us to simplify the present investigation.

1. The law of universal gravitation, as laid down by Newton, is that "Every particle in the universe attracts every other particle, with a force varying directly as the mass of the attracting particle and inversely as the square of the distance between them."

The truth of this law cannot be established by abstract reasoning; but as it is found that the motions of the heavenly bodies, calculated on the assumption of its truth, agree more and more closely with the observed motions as our calculations are more strictly performed, we have every reason to consider the law as an established truth, and to attribute any slight discrepancy between the results of calculation and observation to instrumental errors, to an incomplete analysis, or to our ignorance of the existence of some of the disturbing causes.

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Of the last cause of deviation there is a remarkable instance in the recent discovery of the planet Neptune, for our

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acquaintance with which, as one of the bodies of our system,* we are indebted to the perturbations it produced in the calculated places of the planet Uranus. These perturbations were too great to be attributed wholly to errors of instruments or of calculation; and therefore, either the law of universal gravitation was here at fault, or some unknown body was disturbing the path of the planet. This last supposition, in the powerful hands of Messrs. Adams and Le Verrier, led to the detection of Neptune by solving the difficult inverse problem, viz:-Given the perturbations caused by a body, determine, on the assumption of the truth of Newton's law, the orbit and position of the disturbing body.

Evidence so strong as this forces us to admit the correctness of the assumption, and we shall now consider how this law, combined with the laws of motion, will enable us to investigate the circumstances of the moon's motion, and to assign her position at any time when observation has furnished the requisite data.

2. The problem in its present form would be one of extreme, if not insurmountable difficulty, if we had to take into account simultaneously the actions of the earth, sun, planets, &c. on the moon; but fortunately the earth's attraction, on account of its proximity, is much greater than the disturbing force of the sun or of any planet;-these disturbing forces being so small compared with the absolute force of the earth, that the squares and products of the effects they produce may be neglected, except in extreme

* It had been seen by Dr. Lamont at Munich, one year before its being known to be a planet. "Solar System, by J. R. Hind."

Since the sun attracts both the earth and moon, it is clear that its effects on the moon's motion relatively to the earth or the disturbing force will not be the same as the absolute force on either body. This will be fully investigated in Arts. (9) and (23).

cases: and there is a principle, called the "principle of the superposition of small motions," which shows that in such a case the disturbing forces may be considered separately, and the algebraic sum of the disturbances so obtained will be the same as the disturbance due to the simultaneous action of all the forces.

Principle of Superposition of Small Motions.

3. Let a particle be moving under the action of any number of forces some of which

are very small, and let A be the

position of the particle at any

instant. Let two of these small forces m,, m, be omitted, and suppose the path of the particle under the action of the remaining forces to be AP in any given time.

Let AP, be the path which would have been described in the same time if m, also had acted; AP, differing very slightly from AP, and PP, being the disturbance.

Similarly, if m2 instead of m, had acted, suppose AP, to have represented the disturbed path (AP, AP,, AP, are not necessarily in the same plane, nor even plane curves), PP, being the disturbance.

Lastly, let AQ be the actual path of the body when both m, and m, act. Join PQ.

Now, since the path AP, very nearly coincides with AP, the disturbances PQ and PP, produced in them by the action of the same small force m, will be very nearly parallel; and will differ in magnitude by a quantity which can be only a small fraction of either disturbance and which may be neglected compared with the original path AP. Therefore PQ is parallel and equal to PP.

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Hence the projection of the whole disturbance PQ on any straight line, being equal to the algebraical sum of the pro

jections of PP, and P, Q, will be equal to the algebraical sum of the projections of the separate disturbances PP, PP,

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Now if there are three small disturbing forces m1, m2, m we may consider the joint action of the two m2, m, as one small disturbing force; therefore, by what precedes, the total disturbance along any axis will be the sum of the separate disturbances of m, and of the system m,, m,; but this last is the sum of the separate disturbances of m2 and m2: therefore the whole disturbance equals the sum of the three separate disturbances.

This reasoning can evidently be extended to any number of forces; and if x, y, z be the coordinates of the disturbed particle, (x, y, z) any function of x, y, z; the disturbance produced in 4 (x, y, z) will be

where Sx=8x+x+... sum of disturbances along axis of x

or total disturbance equals sum of separate disturbances, which establishes the principle.

4. Since (x, y, z) may be the radius vector, or the latitude or longitude of the disturbed body, it follows that the total disturbance in any of these elements is the sum of the partial disturbances.

Therefore in determining the motion of a secondary relative to its primary, as in the present case of the moon about the earth, where the disturbing effects produced by the sun and planets are small, we may consider them one at a time, and hence the famous problem of the Three Bodies.

The planets being small and distant, their effect on the motion of the moon will not be of sufficient intensity to affect the order of approximation to which it is intended to carry the solution in the following pages, and our problem is reduced to the consideration of the three bodies, the sun, earth, and moon, acting on one another according to the law of universal gravitation.

5. But we must still prove another proposition, without which the problem would scarcely, though reduced to three bodies, be less complicated than in its most general form.

Newton's law refers to particles, whereas the sun, earth, and moon are large spherical bodies, and it becomes necessary to examine the mutual action of such bodies. Now, it happens that with this law of force, the attraction of one sphere on another can be correctly obtained, and leaves the question in exactly the same state as if they were particles. (Princip. lib. I. prop. 75.)

Attractions of Spherical Bodies.

6. Let P be a particle situated at a distance OP= a from the centre of a uniform attract

ing sphere whose density is p and radius OA=c. a>c, the particle being without the sphere.

Let the whole sphere be divided into circular laminæ by planes perpendicular to OP. Let SQ be

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one of these. PS=x, PQ=z, and thickness of lamina = dx.