GRAVITATION. 



GRAVITATION. 



494 



(98.) IV. We now proceed with the disturbances dependent on the 

 eccentricity : and first with the motion of the moon's perigee. In 

 the first place, suppose that the perigee is on the game side as the 

 sun. While the moon is near BJ, jig. 31, that is, near perigee, the 



Bi 



fig- 31. ,-- '-^ 



c B V y D3 



Bi 



disturbing force is directed from A ; and consequently by (51) the 

 line of apses regresses. While the moon is near B,, that is, near 

 apogee, the disturbing force is also directed from A, and consequently 

 by (54) the line of apses progresses. The question then now is, 

 which is the greater, the regress, when the moon is near By or the 

 progress, when it is near B 3 ? To answer this we will remark, that 

 if the disturbing force directed from A were inversely proportional 

 to the square of the distance (and consequently less at B, than at B,), 

 it would amount to exactly the same as if the attraction of A were 

 altered in a given proportion ; * and in that case B would describe 

 round A an ellipse, whose line of apses was invariable ; or the pro- 

 gression produced at B 3 would be equal to the regression produced 

 at B r But in fact the disturbing force at B 3 is to that at B, in the same 

 proportion as A B, to A B,, by (82) ; and therefore the disturbing 

 force at B 3 is greater than that at B,, and consequently much greater 

 than that which would produce a progression equal to the regression 

 produced at B, ; and therefore the effects of the disturbing force 

 at B S predominate, and the line of apses progresses. The disturbing 

 force directed to A in the neighbourhood of B a and B, scarcely pro- 

 duces any effect, as on one side of each of those points the effect is 

 of one kind, and on the other side it is of the opposite kind (55). 



(99.) The disturbing force directed from A, though the only one 

 at B, and B 3 , is not however the only one in the neighbourhood of 

 Bj and B,. While the moon is approaching to B,, the force perpen- 

 dicular to the radius vector accelerates the moon, and therefore, 

 by (65), an B, is the place of perigee, the line of apses regresses : when 

 the moon has passed B, the force retards the moon, and therefore, by 

 (66), the line of apses still regresses. But when the moon is approaching* 

 B,, the force perpendicular to the radius vector accelerates the moon ; 

 and therefore, by (65) and (66), as B 3 is the plice of apogee, the line of 

 apses progresses ; when the moon has passed B 3 , the force retards the 

 moon, and the line of apses still progresses. The question now is, 

 whether the progression produced by the force perpendicular to the 

 radius vector near B, will or will not exceed the regression produced 

 near B, ? To answer this we must observe, that the rate of this 

 progress or regress depends entirely upon the proportion t which the 



one case as much as in the other, depends entirely upon the difference of the 

 forces in the actual case, from the forces, if the moon were not disturbed. 



Fig. 80. 



* The reasoning in the text may be more fully stated thus : If, with the 

 original attractive force of the earth there be combined another force, directed 

 from the earth, and always bearing the same proportion to the earth's original 

 attraction, this combined force may be considered in two ways : 1st, As a 

 smaller attraction, always proportional to the original attraction, or inversely 

 proportional to the square of the distance. 2nd, As the original attraction, 

 with a fore* superadded, which may be treated as a disturbing force. The 

 result of the first mode of consideration will be, that the moon will describe an 

 ellipse, whose line of apses does not more. The result of the second mode of 

 consideration will be, that the instantaneous ellipse (in which the moon would 

 proceed to move, if the additional force should cease) will have its line of apses 

 regressing, while the moon is near perigee, and progressing while she is near 

 apogee. There is however no incongruity between the immobility of the line of 

 apses in the first mode of consideration, and the progress or regress in the 

 second ; because the line of apses of the instantaneous ellipse in the second case 

 U an imaginary line, determined by supposing the disturbing force to cease, and 

 the moon to man undisturbed. At the apses howerer the line of apses must be 

 the same in both methods of consideration ; since, whether the disturbing force 

 cease or not, the perpendicularity of the direction of the motion to the radius 

 rector determines the place of an apse. Consequently, while the moon moves 

 from one apse to the other, the motions of the line of apses in the second mode 

 of consideration must be such as to produce the same effect on the position of 

 the line of apses as in tho first mode of consideration ; that is, they must not 

 hare altered its place ; and hence the progression at one time must be cxactl\ 

 quarto the regression at the other time. 



t Kuppcte, for facility of conception, that the force perpendicular to the radius 

 reetor acts in only one place in each quadrant between syzygies and quadratures. 

 The portions of the orbit which arc bisected by the line of syzygies will be 

 described with greater velocity in consequence of this disturbance (abstracting 

 all other causes) than the other portions. Now the curvature of any part of at 

 orbit does not depend on the central force simply, or on the velocity, but on the 

 relation between them ; so that the same curve may be described either by 

 leaving the central force unaltered and increasing the velocity in a girei 

 proportion, or br diminishing the central force In a corresponding proportion, 



'elocity produced by the disturbing force bears to the velocity of tho 

 moon ; and since from B., to B D , and from B 3 to B 4 , the disturbing force 

 s greater than that from B, to B W and from B L to B.,, and acts for a 

 .onger time (as by the law of equable description of areas, the moon is 

 onger moving^ from B., to B 3 and B 4 than from B 4 to B, and B a ), and 

 since the moon's velocity in passing through B 2 , B 3 , B 4 , is less than her 

 velocity in passing through B 4 , s v B Q , it follows that the effect in passing 

 ;hrough B 2 , B 3 , B 4 , is much greater than that in passing through B 4 , B^ 

 and B 2 . Consequently, the effect of this force also is to make the line 

 of apses progress. 



(100.) On the whole, therefore, when the perigee is turned towards 

 the sun, the line of apses progresses rapidly: And the same reasoning 

 applies in every respect when the perigee is turned from the sun. 



(101.) In the second place, suppose that the line of apses is per- 

 pendicular to the line joining the earth and sun. The disturbing force 

 at both apses is now directed to the earth, and consequently by (50) 

 and (53), while the moon is near perigee, the disturbing force causes 

 the line of apses to progress, and while the moon is near apogee the 

 disturbing force causes the line of apses to regress. Here, as in the 

 last article, the effects at perigee and at apogee would balance if the 

 disturbing 

 i inn c- from 

 to the distance i 



the disturbing force, while the moon is at apogee, preponderates over 

 the other ; and therefore the force directed to the centre causes the 

 line of apses to regress. 



ill'-.) We must also consider the force perpendicular to the radius 

 vector. In this instance that force retards the moon while she is 

 approaching to each apse, and accelerates her as she recedes from it. 

 The effect is, that when the moun is near perigee the force causes the 

 line of ap >es to progress, and when near apogee it causes the line of 

 apses to regress (65) and (66). The latter is found to preponderate, 

 by the same reasoning as that in (99). From the effect, then, of both 

 causes, the line of apses regresses rapidly in this position of the line of 

 apses. 



(103.) It is important to observe here, that the motion of the line 

 of apses would not, as in (56), be greater if the excentricity of the orbit 

 were smaller ; for, though the motion of the line of apses is greater in 

 j'irujiortion to the farce which causes it when the excentricity is smaller, 

 yet, in the present instance, the force which causes it is itself propor- 

 tional to the excentricity (inasmuch as it is the difference of the forces 

 at perigee and apogee, which would be equal if there were no excen- 

 tricity) : so that if the excentricity were made less, the force which 

 causes the motion of the line of apses would also be made less, and the 

 motion of the line of apses would be nearly the same as before. 



(104.) It appears, then, that when the line of apses passes through 

 the sun, the disturbing force causes that line to progress ; when the 

 earth has moved round the sun, or the sun has appeared to move 

 round the earth, so far that the line of apses is perpendicular to the 

 line joining the sun and the earth, the line of apses regresses from the 

 effect of the disturbing force ; and at some intermediate position it 

 may easily be imagined that the force produces no effect on it. It 

 becomes now a matter of great interest to inquire, whether, upon the 

 whole, the progression exceeds the regression. Now, the force perpen- 

 dicular to the radius vector, considered in (99), is almost exactly equal 

 to that considered in (102) ; so that the progression produced by that 

 force when the line of apses passes through the sun, is almost exactly 

 equal to the regression which it produces when the line of apses is 

 perpendicular to the line joining the earth and sun ; and this force 

 may therefore be considered as producing no effect (except indirectly, 

 as will be hereafter mentioned). But the force in the direction of the 

 radius vector, tending from the earth in (98), is, as we have mentioned 

 in (80), almost exactly double of that tending to the earth in (101), 

 and therefore its effect predominates ; and therefore, on the whole, the 

 line of apses progresses. In fact, the progress, when the line of apses 

 paesei through the sun, is about 11 in each revolution of the moon; 

 the regress, when the line of apses is perpendicular to the line joining 

 the earth and sun, is about 9 in each revolution of the moon. 



(105.) The progression of the line of apses of the moon is considerably 



and leaving the velocity unaltered. Consequently, in the case before us, the 

 same curve will bo described as if, without alteration of velocity, the central 

 force were diminished, while the moon passed through the portions bisected by 

 the line of syzygies. If now the imaginary diminution of central force were in 

 the same proportion (that is, if the real increase of velocity were in the same 

 proportion) at both syzygies, which here coincide with the apses, the regression 

 of the line of apses produced at perigee would be equal to the progression 

 produced at apogee. But the increase of velocity produced by the force 

 perpendicular to the radius vector near apogee is much greater than that near 

 perigee. First, because the force is greater in proportion to the distance. 

 Second, because the time of describing a given small angle is greater in pro- 

 pottion to the square of the distance ; so that the acceleration produced while 

 the moon passes through a given angle is proportional to the cube of the dis- 

 tance. Third, because the velocity, which i increased by this accelciutiuii, i< 

 inversely proportional to the distance ; so that the ratio in uliicli the velocity is 

 Increased is proportional to the fourth power of the distance. The effect at tho 

 greater distance therefore predominates over that at the smaller distance ; and 

 therefore, on the whole, the force perpendicular to the radius vector produces 

 an effect similar to its apogeal effect ; that it, it causes the line of upcs to 

 progress. 



