SECT, ir.] PEOBLEM OF THE THEEE BODIES. 1.3 



Were the planets attracted by the sun only, they would 

 always move in ellipses, invariable in form and position; 

 and because his action is proportional to his mass, which is 

 much larger than that of all the planets put together, the 

 elliptical is the nearest approximation to their true motions. 

 The true motions of the planets are extremely complicated, 

 in consequence of their mutual attraction ; so that they do 

 not move in any known or symmetrical curve, but in paths 

 now approaching to, now receding from, the elliptical form ; 

 and their radii vectores do not describe areas or spaces 

 exactly proportional to the time, so that the areas become a 

 test of disturbing forces. 



To determine the motion of each body, when disturbed by 

 all the rest, is beyond the power of analysis. It is therefore 

 necessary to estimate the disturbing action of one planet at 

 a time, whence the celebrated problem of the three bodies, 

 originally applied to the moon, the earth, and the sun ; 

 namely, the masses being given of three bodies projected 

 from three given points, with velocities given both in quan- 

 tity and direction ; and, supposing the bodies to gravitate to 

 one another with forces that are directly as their masses, 

 and inversely as the squares of the distances, to find the lines 

 described by these bodies, and their positions at any given 

 instant : or, in other words, to determine the path of a ce- 

 lestial body when attracted by a second body, and disturbed 

 in its motion round the second body by a third a problem 

 equally applicable to planets, satellites, and comets. 



By this problem the motions of translation of the celestial 

 bodies are determined. It is an extremely difficult one, and 

 would be infinitely more so, if the disturbing action were not 

 very small when compared with the central force ; that is, 

 if the action of the planets on one another were not very 

 small when compared with that of the sun. As the disturb- 

 ing influence of each body may be found separately, it is 

 assumed that the action of the whole system, in disturbing 

 any one planet, is equal to the sum of all the particular dis- 



