801

160801

moon is (0) or 160000 parts of the sun's attraction on the earth, and the disturbing force is parts of the sun's attraction on the earth, which is very little less than the former. The effects of the difference are, however, sensible.

FIG. 21.

B

a A

(80.) IV. Suppose B, fig. 28, to be in that part of its orbit which is at the same distance from C as the distance of A from C. The attraction of C upon the two other bodies, whose distances are equal, will be equal, but not in the same direction. Bb, therefore, will be equal to A a. But since C B is also equal to CA, it is evident that a b will be parallel to A B, and therefore b will be in the line a d. Consequently in this case also the disturbing force will be entirely in the direction of the radius vector; but here, unlike the other cases, the disturbing force is directed towards the central body. The magnitude of the disturbing force bears the same proportion to the whole attraction on A which b d bears to B b, or A B to A C. Thus, in the first numerical instance taken above, the disturbing force in this part of the orbit is

of the attraction on A: and in the second numerical instance, the disturbing force is of the attraction on A. It is important to observe that in both instances the disturbing force, when wholly directed to the centre, is much less than either value

of the disturbing force when wholly directed from the centre in the latter instance it is almost exactly one-half.

(81.) When the disturbing body is distant, the point of the orbit which we have here considered is very nearly that determined by drawing A B perpendicular to CA.

(82.) V. When C is distant (as in the case of the

moon disturbed by the sun), the disturbing forces mentioned in (III.) and (IV.) are nearly proportional to the distance of the moon from the earth. For the force mentioned in (IV.) this is exactly true, whether C be near or distant, because (as we have found) the disturbing force bears the same proportion to the whole attraction on A which A B bears to A C. With regard to the force mentioned in (III.); if we suppose the moon's distance from the earth to be of the sun's distance, the disturbing force when the moon is between the earth and the sun is TOT parts of the sun's attraction on the earth, or nearlyth part. But if we suppose the moon's distance from the earth to beth of the sun's distance, the attraction on the moon (when between the earth and the sun) would be (199) or 9001 parts of the attraction on the earth; the disturbing force, or the difference of attractions on the earth and moon, would be 388, or nearly th part of the 399 nearlyth sun's attraction on the earth. Thus, on doubling the moon's distance from the earth, the disturbing

799

40000

force is nearly doubled: and in the same manner, on altering the distance in any other proportion, we should find that the disturbing force is altered in nearly the same proportion.

(83.) VI. If, while the moon's distance from the earth is not sensibly altered, the earth's distance from the sun is altered, the disturbing force is diminished very nearly in the same ratio in which the cube of the sun's distance is increased. For if the sun's distance is 400 times the moon's distance, and the moon between the earth and the sun, we have seen that the disturbing force is nearlyth part of the sun's attraction on the earth at that distance of the sun. Now, suppose the sun's distance from the earth to be made 800 times the moon's distance, or twice the former distance: the sun's distance from the moon will be 799 times the moon's distance, or 88 parts of the sun's former distance from the earth; the attractions on the earth and moon respectively will be and 190000 parts of the former attraction on the earth: and the disturbing force, or the difference between these, will be or nearly 6th part of the former attraction of the earth. Thus, on doubling the sun's distance, the disturbing force is diminished to th part of its former value; and a similar proposition would be found to be true if the sun's distance were altered in any other proportion.

1599 2553604

99

638401

(84.) VII. Suppose B to have moved from that part of

its orbit where its distance from C is equal to A's

distance from C, towards the part where it is between A and C. Since at the point where B's distance from C is equal to A's distance from C, the disturbing force is in the direction of the radius vector, and directed towards A, and since at the point where B is between A and C, the disturbing force is in the direction of the radius vector, but directed from A, it is plain that there is some situation of B, between these two points, in which there is no disturbing force at all in the direction of the radius vector. On this we shall not at present speak further; but we shall remark that there is a disturbing force perpendicular to the radius vector, at every such intermediate point. This will be easily seen from the second case of fig. 17. On going through the reasoning in that place it will appear that, between the two points that we have mentioned, there is always a disturbing force da e perpendicular to the radius vector, and in the same direction in which the body is going. If now we construct a similar figure for the situation B,

fig. 22, in which B is moving from the point between C and A to the other point whose distance from C is equal to A's distance from C, we shall find that there is a disturbing force da e perpendicular to the radius

vector, in the direction opposite to that in which B is going. If we construct a figure for the situation B in which B is moving from the point of equal distances, to the point where B is on the side of A opposite to C, we shall see that there is a disturbing force perpendicular to the radius vector, in the same direction in which B is going; and in the same manner, for the situation B1, in fig. 17, where B is moving from the point on the side of A opposite C to the next point of equal distances, there is a disturbing force perpendicular to the radius vector, in the direction opposite to that in which B is going.

(85.) The results of all these cases may be collected thus. The disturbing body being exterior to the orbit of the revolving body, there is a disturbing force in the direction of the radius vector only, directed from the central body, at the points where the revolving body is on the same side of the central body as the disturbing body, or on the opposite side (the force in the former case being the greater), and directed to the central body, at each of the places where the distance from the disturbing body is equal to the distance of the central body from the disturbing body. The force directed to the central body at the latter points is, however, much less than the force directed from it at the former. Between the adjacent pairs of these four points there are four other points, at which the disturbing force in the direction of the radius vector is nothing. But while the revolving body is moving from one of the points, where it is on the same side of the central body as the