601 



GRAVITATION. 



GRAVITATION. 



602 



We shall now proceed with the application of these conclusions to 

 Jupiter's first three satellites. 



(135.) The periodic time of Jupiter's first satellite is, 1 day, 18 hours, 

 27 minutes, and 34 seconds ; that of the second satellite is, 3 days, 

 13 hours, 13 minutes, and 42 seconds; that of the third satellite is, 

 7 days, 3 hours, 42 minutes, and 32 seconds. The periodic time of 

 the second satellite exceeds, by a small quantity, double that of the 

 first, so that the preceding investigations apply to the motion of these 

 two satellites. In fact, 275 revolutions of the first satellite are 

 finished in almost exactly the same time as 137 revolutions of the 

 second. If then, at a certain time, these two satellites start from con- 

 junction, they will be in conjunction near the same place at every 

 revolution of the second satellite, or at every second revolution of the 

 first satellite : but the line of conjunction will regress slowly ; and 

 when the first satellite has finished 275 revolutions, or one revolution 

 more than double the number made by the second satellite, they will 

 again be in conjunction in the same place as before, the line of con- 

 junction having regressed till it has again reached the same position ; 

 this takes place in 486 4 days. 



(136.) From the preceding investigation then it appears that, as 

 these orbits have no eccentricity independent of perturbation, they 

 will be elliptic, and the line of apses of each orbit will regress so as to 

 turn completely round in 4864 days ; and that when in conjunction, 

 the first satellite will always be in perijove, and the second satellite 

 will always be in apojove. 



(137.) But the periodic time of the third satellite is almost exactly 

 double that of the second satellite, exceeding the double by a small 

 quantity ; and on this account the orbit of the second satellite will be 

 distorted from the form which otherwise it would have had, by an 

 inequality similar to that just investigated. In a word, the line of 

 conjunction of the second and third satellites will slowly regress, and 

 the orbit of the second satellite will always be compressed on the side 

 next the points of conjunction, and elongated on the opposite side; 

 and the orbit of the third satellite will always be elongated on the 

 side next the points of conjunction, and compressed on the opposite 

 side. 



(138.) Now we come to the most extraordinary part of this theory. 

 We have remarked that 275 revolutions of the first satellite are finished 

 in almost exactly the same time as 137 revolutions of the second ; but 

 it will also be found that 137 revolutions of the second are finished in 

 almost exactly the same time as 68 revolutions of the third : all these 

 revolutions occupying 4804 days. Because 275 exceeds the double of 

 137 by 1, we have inferred that the line of conjunctions of the first 

 and second satellites regresses completely round in 275 revolutions of 

 the first satellite, or in 4864 day 6 - J D like manner, because 137 exceeds 

 the double of 68 by 1, we infer that the line of conjunctions of the 

 second and third satellites regresses completely round in 137 revolu- 

 tions of the second satellite, or in 4864 days. Hence we have this 

 remarkable fact : the regression of the line of conjunction of Ike second 

 and third satellites is exactly at rapid as the reyression of the line of con- 

 junction of the first and second satellites. So accurate is this law, that 

 in the thousands of revolutions of the satellites, which have taken place 

 since they were discovered, not the smallest deviation from it (except 

 what depends upon the elliptic form of the orbit of the third satellite) 

 has ever been discovered. 



(139.) Singular as this may appear, the following law is not less so. 

 The line of conjunction of the second and third satellites always eoinailes 

 trith the line of conjunction of the fnt and second satellites produced 

 batticards, the conjunctions of the second and third satellites always 

 taking place on tJu tide opposite to that on which the conjunctions of the 

 nd second take place. This defines the relative position of the 

 lines of conjunction, which (by the law of last article) is invariable. 

 Like that law it has been found, as far as observation goes, to be 

 accurately true in every revolution since the satellites were discovered. 



(140.) The most striking effect of these laws in the perturbations of 

 the satellites is found in the motions of the second satellite. In con- 

 sequence of the disturbing force of the flint satellite, the orbit of the 

 second satellite will be elongated towards the points of conjunction of 

 the first and second (130), and consequently compressed on the 

 opposite side. In consequence of the disturbing force of the third 

 satellite, the orbit of the second satellite will be compressed on the side 

 next the points of conjunction of the second and third (128). And 

 because the point* of conjunction of the second and third are always 

 opposite to the points of conjunction of the first and second, the place 

 of compression from one cause will always coincide with the place of 

 compression from the other cause; and therefore the orbit of the 

 second satellite will be very much compressed on that side, and con- 

 sequently very much elongated on the other side. The excentricity 

 of toe orbit, depending thus entirely on perturbation, exceeds 

 considerably the excentricity of the orbit of Venus. The inequalities 

 in the motions of the satellites, produced by these eccentricities, were 

 first discovered (from observation) by Bradley about A.I). 1710, and 

 nod from theory by Lagrange, iu 1766. 



(111.) The singularity of these laws, and the accuracy with which 

 they . i, lead us to suppose that they do not depend entirely 



on chance. It seems natural to inquire, whether some reason may not 

 be found in the mutual disturbance of the satellites, for the preserva- 

 tion of such simple relations. Now we are able to show that, 



supposing the satellites put in motion at any one time, nearly iu 

 conformity with these laws, their mutual attraction would always tend 

 to make their motions follow these laws exactly. We shall show this 

 by supposing a small departure from the law, and investigating the 

 nature of the forces which will follow as a consequence of that 

 departure. 



(142.) Suppose, for instance, that the third satellite lags behind the 

 place defined by thia law; that is, suppose that when the second 

 satellite is at the most compressed part of its ellipse, (as produced by 

 the action of the first satellite,) the third satellite is behind that place. 

 The conjunction then of the second and third satellites will happen 

 before reaching the line of apses of the orbit of the second, as produced 

 by the action of the first. Now in the following estimation of the 

 forces which act on the third satellite, and of their variation depending 

 on the variation of- the positions of the lines of conjunction, there is no 

 need to consider the influence which the ellipticity of the orbit of the 

 second as produced by the third, or that of the third as produced by 

 the second, exerts upon the third satellite; because the flattening 

 arising from the action of the third, and the elongation arising from 

 the action of the second, will always be turned towards the place of 

 conjunction of the second and third, and the modification of the action 

 produced by this flattening and elongation will always be the same, 

 whether the lines of conjunction coincide or not. In jig. 37, let c be 



1'ig. S7. 



the perijove of the orbit of the second satellite, (as produced by the 

 action of the 1st satellite alone,) D the point of the orbit of the third, 

 which is in the hue A c produced. If the third satellite is at D when 

 the second is at c, the force produced by the second perpendicular to 

 the radius vector, retards the third before it 'reaches D, and accelerates 

 it after it has passed D, by equal quantities. But if, as in the suppo- 

 sition which we have made, the conjunction takes place in the line 

 A c, D,, the retardation of the third satellite before conjunction is pro- 

 duced by the attraction of the second satellite before it arrives at 

 perijove, when it is near to the orbit of the third satellite, (and there- 

 fore acts powerfully,) and moves slowly, (and therefore acts for a long 

 time) ; while the acceleration after conjunction is produced by the 

 second satellite near its perijove, when it is far from 1;he orbit of the 

 third satellite, (and therefore acts weakly,) and moves rapidly (and 

 therefore acts for a short time). The retardation therefore exceeds 

 the acceleration ; and the consequence is, by (48), that the periodic 

 time of the third satellite is shortened, and therefore its angular motion 

 is quickened ; and therefore, at the next conjunction, it will have gone 

 further forward before the second satellite can come up with it, or the 

 line of conjunction will be nearer to the place of perijove of the second 

 satellite, depending on the action of the first. In the same manner, 

 if we supposed the third satellite moving rather quicker than it ought 

 in conformity with the law, the tendency of the forces would be to 

 accelerate it, to make its periodic time longer, and thus to make its 

 angular motion slower. By the same kind of reasoning it will be seen 

 that there are forces acting on the first satellite, produced by the 

 elliptic inequality which the third impresses on the orbit of the second, 

 tending to accelerate the angular motion of the first satellite in the 

 first case, and to retard it in the second. The same reasoning will 

 also show that both the first and third satellites exert forces on the 

 second, tending to retard its angular motion in the first case, and to 

 accelerate it in the second. All these actions tend to preserve the 

 law : in the first case by making the line of conjunctions of the first 

 and second satellite regress, and that of the second and third progress, 

 till they coincide ; and in the second case, by altering them in the 

 opposite way, till they coincide. 



(143.) Perhaps there is no theoretical permanence of elements on 

 which we can depend with so great certainty, as on the continuance of 

 this law. The greatest and most irregular perturbations of Jupiter or 

 of his satellites, provided they come on gradually, will not alter tlio 

 relation between their motions ; the effect of a resisting medium will 

 not alter it ; though each of these causes would alter the motions of 

 all the satellites ; and though similar causes would wholly destroy tlio 

 conclusions which mathematicians have drawn as to the stability of the 

 solar system, with regard to the elements of the planetary orbits. 

 The physical explanation of this law was first given by Laplace, iu 

 A.D. 1784. 



(144.) We have terminated now the most remarkable part of the 

 theory of these satellites. There are however some other points which 

 are worth attending to, partly for their own sake, and partly as an 

 introduction to the theory of the planets. 



(1 l'i.) The orbit of the third satellite, as we have mentioned, has a 

 small excentricity independent of perturbation. Consequently, when 

 the conjunction with tho second takes place near the independent 



