(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 486 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 486 days. In 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 revolutions of the second satellite, or in 486 days. Hence we have this remarkable fact: the regression of the line of conjunction of the second and third satellites

is exactly as rapid as the regression of the line of conjunction 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 coincides with the line of conjunction of the first and second satellites produced backwards, the conjunctions of the second and third satellites always taking place on the side opposite to that on which the conjunctions of the first and second take place. This defines the relative position of the lines of conjunction, which (by the law of the 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 consequence of the disturbing force of the first 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 points 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 consequently very much elongated on the other side. The excentricity of the 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 excentricities, were first discovered (from observation) by Bradley about A.D. 1740, and first explained from theory by Lagrange, in 1766.

(141.) The singularity of these laws, and the accuracy with which they are followed, 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 disturbances of the satellites, for the preservation of such simple relations. Now we are able to show that, supposing the satellites put in motion at any one time, nearly in 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 this 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 fig. 37, let C be the peri

jove of the orbit of the second satellite (as produced by the action of the first satellite alone), D the point of the orbit of the third which is in the line 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 supposition which we have made, the conjunction takes place in the line A C, D1, the retardation of the third satellite before conjunction is produced by the attraction of the second satellite before it arrives at perijove, when it is near to the orbit of the third satellite (and therefore 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 the 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