S*CT. II. PROBLEM OF THE THREE BODIES. 11 



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 ap- 

 proaching 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 be- 

 come a test of disturbing forces. 



To determine the motion of each body, when dis- 

 turbed by all the rest, is beyond the power of analysis. 

 It is therefore necessary to estimate the disturbing ac- 

 tion 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 quantity and direc- 

 tion ; and, supposing the bodies to gravitate to one an- 

 other with forces that are directly as their masses, and 

 Diversely 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 celestial 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 dis- 

 turbing action were not very small when compared with 

 the central force ; that is, if the action of the planets on 

 one another were not veiy small when compared with 

 that of the sun. As the disturbing 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 disturbances it 

 experiences, on the general mechanical principle, that 

 the sum of any number of small oscillations is nearly 

 equal to their simultaneous and joint effect. 



On account of the reciprocal action of matter, the 

 stability of the system depends upon the intensity of the 



