489 



GRAVITATION. 



GRAVITATION. 



490 



dicular to the radius vector, in the same direction in which B is 

 going ; and in the same manner, for the situation B, in fiq. 1 7, 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 dis- 

 turbing 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 disturbing body, or on the opposite side, to one 

 of the equidistant points, there is always a disturbing force perpen- 

 dicular to the radius vector tending to retard it ; and while it is 

 moving from one of the equidistant points to one of the points on the 

 same side of the central body as the disturbing body, or the opposite, 

 there is a disturbing force perpendicular to the radius vector tending 

 to accelerate it. 



(86.) VIII. Now, let the disturbing body be supposed interior to the 

 orbit of the revolving body (as, for instance, when Venus disturbs 

 the motion of the earth). If B is in the situation B,, fg. 23, the 



Fig. 23. 



attraction of c draws A strongly towards B,, and B, strongly 

 towards A, and, therefore, there is a very powerful disturbing force 

 drawing B, towards A. If B is in the situation B 3 , the attraction of 

 c draws A strongly from B 3 , and draws B 3 feebly towards A ; there- 

 fore, there is a small disturbing force drawing B 3 from A. At some 

 intermediate points the disturbing force in the direction of the 

 radius vector is nothing. With regard to the disturbing force per- 

 pendicular to the radius vector : if A c is greater than J A B,, it will 

 be possible to find two points, B, and B,, whose distance from c is 

 equal to the distance of A from c, and there the disturbing force 

 perpendicular to the radius vector is nothing (or the whole disturb- 

 ing force is in the direction of the radius vector). While B moves 

 from the position B, to B,, it will be seen by such reasoning as that 

 of (75) and (84), that the disturbing force, perpendicular to the 

 radius vector, retards B's motion ; while B moves from B a to BJ, it 

 accelerates B'S motion ; while B~ moves from B, to B, it retards B'S 

 motion ; and while B moves from B, to B,, it accelerates B'S motion. 

 But if AC is less than 4 AB,, there are no such points, B, B,,as we 

 have spoken of ; and the disturbing force perpendicular to the radius 

 vector, accelerates B as it moves from B, to B 3 , and retards B as it 

 moves from B 3 to B,. 



We shall now proceed to apply these general principles to particular 

 cases. 



SECTION V. Lunar Theory. 



(87.) The distinguishing feature in the Lunar Theory is the general 

 simplicity occasioned by the great distance of the disturbing body (the 

 nun alone producing any sensible disturbance), in proportion to the 

 moon's distance from the earth. The magnitude of the disturbing 

 body renders these disturbances very much more conspicuous than any 

 others in the solar system ; and, on this account, as well as for the 

 accuracy with which they can be observed, these disturbances have, 

 since the invention of the Theory of Gravitation, been considered the 

 best tests of the truth of the theory. 



Some of the disturbances are independent of the excentricity of the 

 moon's orbit ; others depend, in a very remarkable manner, upon the 

 excentricity. We shall commence with the former. 



(88.) The general nature of the disturbing force on the moon may 

 b thug stated. (See (77) to (86).) When the moon is either at the 

 point between the earth and sun, or at that opposite to the sun (both 

 which points are called syzygies), the force is entirely in the direction 

 of the radius vector, and directed from the earth. When the moon is 

 (very nearly) in the situations at which the radius vector is perpen- 

 dicular to the line joining the earth and sun (both which points are 

 called quadratures), the force is entirely in the direction of the radius 

 vector, and directed to the earth. At certain intermediate points there 

 a no disturbing force in the direction of the radius vector. Except at 



s yzygies and quadratures, there is always a force perpendicular to the 

 radius vector, such as to retard the moon while she goes from syzygy to 

 quadrature, and to accelerate her while she goes from quadrature to 

 syzygy. 



(89.) I. As the disturbing force, in the direction of the radius vector 

 directed from the earth, is greater than that directed to the earth, 

 we may consider that, iipon the whole, the effect of the disturbing 

 force is to diminish the earth's attraction. Thus the moon's mean 

 distance from the earth is less (see (46) ) than it would have been 

 with the same periodic time, if the sun had not disturbed it. The 

 force perpendicular to the radius vector sometimes accelerates 

 the moon, and sometimes retards it, and, therefore, produces no 

 permanent effect. 



(90.) II. But the sun's distance from the earth is subject to altera- 

 tion, because the earth revolves in an elliptic orbit round the sun. 

 Now, we have seen (83) that the magnitude of the disturbing force 

 is inversely proportional to the cube of the sun's distance ; and, con- 

 sequently, it is sensibly greater when the earth is at perihelion than 

 when at aphelion. Therefore, while the earth moves from perihelion 

 to aphelion, the disturbing force is continually diminishing^ and 

 while it moves from aphelion to perihelion, the disturbing force is 

 constantly increasing. Referring then to (47) it will be seen, that 

 in the former of these times the moon's orbit is gradually diminish- 

 ing, and that in the latter it is gradually enlarging. And though 

 this alteration is not great (the whole variation of dimensions, from 

 . greatest to least, being less than 5^, the effect on the angular 

 motion (see (49) ) is very considerable; the anguiar velocity 

 becoming quicker in the former time and slower in the latter ; so 

 that while the earth moves from perihelion to aphelion, the moon's 

 angular motion is constantly becoming quicker, and while the earth 

 moves from aphelion to perihelion the moon's angular motion is 

 constantly becoming slower. Now, if the moon's mean motion is 

 determined by comparing two places observed at the interval of 

 many years, the angular motion so found is a mean between the 

 greatest and least. Therefore, when the earth is at perihelion, the 

 moon's angular motion is slower than its mean motion ; and when 

 the earth is at aphelion, the moon's angular motion is quicker than 

 its mean motion. Consequently, while the earth is going from 

 perihelion to aphelion, the moon's true place is always behind its 

 mean place (as during the first half of that period the moon's true 

 place is dropping behind the mean place, and during the latter half 

 is gaining again the quantity which it had dropped behind) ; and 

 while the earth is going from aphelion to perihelion, the moon's 

 true place is always before its mean place. This inequality is called 

 the moon's annual equation ; it was discovered by Tycho Brah<5 

 from observation, about 1590 ; and its greatest value is about 

 10', by which the true place is sometimes before and sometimes 

 behind the mean place. 



(91.) III. The disturbances which are periodical in every revolution of 

 the moon, and are independent of excentricity, may thus be investi- 

 gated. Suppose the sun to stand still for a few revolutions of the 

 moon (or rather suppose the earth to be stationary), and let us 

 inquire in what kind of orbit, symmetrical on opposite sides, the sun 

 can move. It cannot move in a circle : for the force perpendicular 

 to the radius vector retards the moon as it goes from B, to B a , Jig. 24, 



Fig. 24, 



and its velocity is therefore less at B a than at B,, and on this account 

 (supposing the force directed to A at B a equal to the force directed 

 to A at B,), the orbit would be more curved at B a than at B,. But 

 the force directed to A at B, is much greater than at B, (see (88) ) ; 

 and on this account the orbit would be still more curved at B.J than 

 at BJ ; whereas, in a circle, the curvature is everywhere the same. 

 The orbit cannot therefore be circular. Neither can it be an oval 

 with the earth in its centre, and with its longer axis passing through 

 the sun, as fg. 25 ; for the velocity being small at B a (in consequence 



B 



Fig. 25. C fjj~\ 



vlv 3 



of the disturbing force perpendicular to the radius vector having 

 retarded it) while the earth's attraction is great (in consequence of 

 the nearness of B 2 ), and increased by the disturbing force in the 

 radius vector directed towards the earth, the curvature at B 2 ought 

 to be much greater than at B,, where the velocity is great, the moon 

 far off, and the disturbing force directed from the earth. But, on 

 the contrary, the curvature at B 2 is much less than at B, ; therefore, 

 this form of orbit is not the true one. But if the orbit be supposed 

 to be oval, with its shorter axis directed towards the sun, as in 



