*-: 



GRAVITATION. 



GRAVITATION. 



(75.) Both the magnitude nnd the direction of thii fore* ore con- 

 tinually varying, Mid we mutt, if possible, find a convenient way of 

 representing it. We Khali have recoune hare to the " compositon of 



Flf. 11. 



motion." In Jtg. 18, if d b represent the anace through which a force 

 ha* drawn a body in a certain time, the same effect may be produced 

 by two force* of which one would in the came time draw the body 

 from d to e, and the other would in the same time draw the body from 

 < to 6. And thin i* true whatever be the directions and length* of 

 d t and b, provided that with d b they form a triangle. To accom- 

 modate the investigations of this Section to those of Section 111., we 

 will suppose il e |ierpendiculnr to the radius vector, and e b parallel to 

 the radius vector. In .fiy. 17, draw d e perpendicular to A B or a d, 

 and t b parallel to A B or ad; and now we can say : the disturbing 

 fore* produced by the attraction of c is a force represented by d t 

 perpendicular to the radius vector, and a force represented by t b in 

 the direction of the radius vector. 



(76.) We now want nothing but estimations of the magnitudes of 

 these forces in order to apply the investigations of Section III. For 

 the present we shall content ourselves with pointing out some of the 

 most interesting cases. 



(77.) I. Let the disturbing body be exterior to the orbit of the 



disturbed body: (this applies to the disturbance of the moon's 



motion produced by the sun's attraction, the disturbance of the 



earth's motion by Jupiter's attraction, the disturbance of the motion 



Fig. 19. C b d B a A 



of Venus by the earth's attraction, Ac. :) and first, let the revolving 

 body B be between the disturbing body c and the central body A 

 (as in Jig. 19.). If the attraction of c will in a certain time draw A 

 to a, it will in the same time draw B to 6, where B 6 is much greater 

 than A a. Take B d equal to A a, then d b is the effect of the 

 disturbing force, which tends to draw B further from A. In this 

 case then, the disturbing force is entirely in the direction of the 

 radius vector, and directed from the central body. This is the 

 greatest disturbing force that can be produced by c. 

 ^78.) II. Let CAB (fy. 20) be in the same straight line, but let B be 

 Fig. 20. C a A d B 



T 



on the side of A, opposite to c. In this case B 6 is less than A a ; 

 and if B d is taken equal to A a, the disturbing force represented by 

 db will be entirely in the direction of the radius vector, and directed 

 from the central body. This case is particularly deserving of the 

 reader's consideration, as the effectual ditturbing force is exactly 

 opposite to the attraction which C actually exerts upon B. 

 (79.) III. The disturbing force in the case represented in fg. 19, is 

 much greater than that in the case of- fig. 20, except c be very 

 distant. Thus, suppose A B to be half of A c. In the first case, the 

 attraction upon B (by the law of gravitation) is four times as great 

 as the attraction upon A, and therefore the disturbing force (which 

 is the difference of the forces on A and B) is three times as great as 

 the attraction upon A. In the second case, the distance of B is j of 

 the distance of A, and therefore the attraction upon B is J of the 

 attraction upon A, and .the disturbing force is J of the attraction 

 upon A. The disturbing force in the first case is, therefore, greater 

 than in the second case, in the proportion of 3 to j, or 27 to 5. 

 This remark applies to nearly all the cases of planetary disturbance 

 where the disturbing planet is exterior to the orbit of the disturbed 

 planet, the ratio between these distances from the sun being a ratio 

 of not very great inequality. But it scarcely applies to the moon. 

 For the sun's distance from the earth is nearly 400 times the 

 moon'* distance : consequently when the moon is between the sun 

 and the earth, the attraction of the sun on the moon is ((,')* * tne 

 atti action of the sun on the earth, or {y_gy parts of the sun's attrac- 

 tion on the earth, and the disturbing force therefore is ^iSSn part* of 

 the sun's attraction on the earth ; but when the moon is on that side 

 farthest from the sun, the sun's attraction on the moon is ( \'J'J : or 

 it iJiV parts ' * ne *un's attraction on the earth, and the disturbing 

 force i ,i';\>< parts of the sun's attraction on the earth, which in very 

 little li-M than the former. The effects of the difference are, bow- 

 ever, sensible. 



equal to AH. But since CB i* also equal to c A, it is evident that 06 

 will be parallel to A B, and therefore b will be in the line a d. i >n 

 sequent!}- in this cue also the disturbing force will be entirely in 

 the direction of the radius vector ; but here, unlike the other case., 

 the disturbing force is directed totrardt 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 6, or A B to A c. Thus, in 

 the first numerical instance taken above, the disturbing force in 

 this part of the orbit is i of the attraction on A : and in the second 

 numerical instance, the disturbing force is jfc of the attraction on A. 

 mi|M>rtant to observe that in both instances the disturbing 

 wl..-n wholly directed to the centre, is much lew than . 

 value of the disturbing force when wholly directed /ro the 

 centre : in the latter instance it is almost exactly one-half. 

 81. When the disturbing body is distant, the jwint of tin- orl.it 

 which we have here considered is very nearly that determined by 

 drawing A B perpendicular to C A. 



(8'J.) 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 b 

 near or distant, because (as we have found) the disturbing force 

 bears the same proportion to the whole attraction on A which A B 

 bean to AC. With regard to he force mentioned in (III.) ; if we 

 suppose the moon's distance from the earth to be ,4, of the sun 1 * 

 distance, the disturbing force when the moon is between the earth 

 an.l the sun is T #3, T parts of the sun's attraction on the earth, or 

 nearly ,foth part. But if we suppose the moon's distance from HM 

 wirth to be ,foth of the sun's distance, the attraction on the moon 

 i w h.-n between the earth and the sun) would be (JJj) or 

 of the attraction on the earth ; the disturbing force or the 

 of attractions on the earth and moon, would be jgfo, or nearly , 

 part of the sun's attraction on the earth. Thus, on doubling the 

 moon's distance from the earth, the disturbing 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 nearly .fcth part 

 of the sun's attraction on the caith at thai distance of the tun. 

 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 

 Jjjg part* of the sun's former distance from the earth ; the attraction* 

 on the earth and moon respectively will be i and JffiS? P"" 4 " of the 

 former attraction on the earth : and the disturbing force, or the 

 difference between these, will be M&s, or nearly ,&jth part of the 

 former attraction of the earth. Thus, on doubling the sun's dis- 

 ' tanee, the disturbing force is diminished to Jth 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. 

 (84 VII. Suppose B to have moved from that part of its orbit where 

 its distance from o is equal to A'B 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 tincariU 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 frum A, it in pliin 

 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 po-pendiouhu' to the radius 

 vector, at every such intermediate point. This will be easily seen 

 from the second case of fg. 17. On going through the reasoning in 

 that place it will appear that, between the two points that we nave 

 mentioned, there is always a disturbing force d t 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,,.% 22, i" 



(80.) IV. Suppose B, fg. 21, to be in that part of it* orbit which is at 

 the *ame distance from c a* the distance of A from c. The attrac- 

 tion of c upon the to other bodies, whose distances are equal, will 

 1 e equal, but not in th* same direction. B 6, therefore, will be 



Fig. 22. 



which B hi 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 rf,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 hall see that there is a disturbing force perpcn- 



