LAPLACE. 145 



of tbe conception and the ultimate result are liere equally worthy of 

 admiration. * 



* The origin of this famous inoquality may be best understood by reference to the mode 

 in which the disturbing forces operate. Let P Q R, P' Q' R' represent the orbits of Jupiter 

 and Saturn and let us suppose, for the sake of illus- , 



tration, that they are both situate in the same plane. r'y^ — ' ~"^\ 



Let the planets be in conjunction at P, P', and let >/\. \ ^^\N- 



them both be revolving around the sun S, in the / \ \ j^ \ 



direction represented by the arrows. Assuming that / /?'T' ~"^^\V ^ 



the mean motion of Jupiter is to that of Saturn ex- / / \\ \ \ 



actly in the proportion of five to two, it follows that t'[ ^/ "'\ Xp P 



when Jupiter has completed one revolution, Saturn I I / ~|^^ \'^ 



will have advanced through two-fifths of a revolu- \ I // \ \ 



tion. Similarly, when Jupiter has completed a revo- \ \ / •' J I 



lution and a half, Saturn will have eflected three- \ y\/ y / 



fifths of a revolution. Hence, when Jupiter arrives ^\/ /^^ / 



at T, Saturn will be a little in advance of T'. Let ^^^^"--^^ ^-^-^^^ 



us suppose that the two planets come again into con- ^ 



junction at Q, Q'. It is plain that while Jupiter has completed one revolution, and advanced 

 through the angle P S Q, (measured in the direction of the arrow, ) Saturn has simply described 

 around S the angle P' S' Q'. Hence the excess of the angle described around S, by Jupiten 

 over the angle similarly described by Saturn, will amount to one complete revolution, or 

 360°. But since the mean motions of the two planets are in the proportion of five to two, 

 the angles described by them around S in any given time will be in the same proportion, and 

 therefore the ex'jess of the angle described by Jupiter over that described by Saturn will be 

 to the angle described by Saturn in the proportion of three to two. But we have just found! 

 that the excess of these two angles in the present case amounts to 360°, and the angle de- 

 scribed by Saturn is represented by P' S' Q' ; consequently 360° is to the angle P' S' Q' in 

 the proportion of three to two ; in other words, P' S' Q' is equal to two-thirds of the circum 

 ference, or 240°. lu the same w^ay it may be shown that the two planets will come into con- 

 junction again at R, when Saturn has described another arc of 240°, Finally, when Saturu 

 has advanced through a third arc of 240°, the two planets will come into conjunction at P, P', 

 the points whence they originally set out ; and the two succeeding conjunctions will also mani- 

 festly occur at Q, Q' and R, R'. Thus wo sec that the conjunctions will always occur in three 

 given points of the orbit of each planet situate at angular distances of 120° from each other. 

 It is also obvious that, during the interval which elapses between the occurrence of two con- 

 junctions in the same points of the orbits, and which includes three synodic revolutions of the 

 planets, Jupiter will have accomplished five revolutions around the sun, and Saturn will have 

 accomplished two revolutions. Now, if the orbits of both planets were perfectly circular, the 

 retarding and accelerating effects of the disturbing force of either planet would neutralize each 

 other in the course of a synodic revolution, and therefore both planets would return to the same 

 condition at each successive conjunction. But in consequence of the ellipticity of the orbits, the 

 retarding effect of the disturbing force is manifestly no longer exactly compensated by the 

 accelerative effect, and hence, at the close of each synodic revolution, there remains a min- 

 ute outstanding alteration in the movement of each planet. A similar effect will be pro- 

 duced at each of the three points of conjunction, and as the perturbations which thus ensue- 

 do not generally compensate each other, there will remain a minute outstanding perturbation 

 as the result of every three conjunctions. The efi'ect produced being of the same kind (whether 

 tending to accelerate or retard the movement of the planet) for every such triple conjunction, it 

 is plain that the action of the disturbing forces would ultimately lead to a serious derangement 

 of the movements of both planets. All this is founded on the supposition that the mean 

 motions of the two planets are to each other as two to five, but in reality this relation does 

 not exactly hold. In fact while Jupiter requires 21 ,GG3 days to accomplish five revolutions, 

 Saturn effects two revolutions -in 21,518 days. Hence when Jupiter after completing his 

 fifth revolution arrives at P, Saturn will have advanced a little beyond P', and the con- 



10 s 



