SCIENCE. 



LN. S. Vol. XVII. No. 418. 



the other planets. By means of this 

 theory, and the mathematical machinery 

 given by Lagrange, which can be applied 

 to a great variety of questions, the observs/- 

 tions of the planets can be satisfied over 

 long intervals of time. When this theory 

 of the motions was carried out a century 

 ago it appeared that the great problem of 

 planetary motion was near a complete so- 

 lution. But this solution depends on the 

 use of series, which undergo integrations 

 that may introduce small divisors. An 

 examination of these series by Hansen, 

 Poincar.e and others indicates that some 

 of them are not convergent. Hence the 

 conclusions formerly drawn about the 

 stability of our solar system are not trust- 

 worthy, and must be held in abeyance. 

 But looking at the construction of our sys- 

 tem, and considering the manner in which 

 it was probably evolved, it appears to be 

 stable. However the mathematical proof 

 is wanting. In finding the general inte- 

 grals of the motions of n bodies, the as- 

 sumption that the bodies are particles gets 

 rid of the motions of rotation. These 

 motions are peculiar to each body, and are 

 left for special consideration. In the case 

 of the earth this motion is very important, 

 since the reckoning of time, one of our 

 fundamental conceptions, depends on this 

 motion. Among the ten general integrals 

 that can be found six belong to the pro- 

 gressive motion of the system of bodies. 

 They show that the center of gravity of 

 the system moves in a right line, and with 

 uniform velocity. Accurate observations 

 of the stars now extend over a century 

 and a half, and we are beginning to see 

 this result by the motion of our sun 

 through space. So far the motion appears 

 to be rectilinear and uniform, or the action 

 of the stars is without influence. This is 

 a matter that will be developed in the 

 future. Three of the other general inte- 



grals belong to the theory of areas, and 

 Laplace has drawn from them his theory 

 of the invariable plane of the system. The 

 remaining integral gives the equation of 

 living force. The question of relative mo- 

 tion remains, and is the problem of the- 

 oretical astronomy. This has given rise 

 to many beautiful mathematical investi- 

 gations, and developments into series. But 

 the modern researches have shown that we 

 are not sure of our theoretical results ob- 

 tained in this way, and we are thrown 

 back on empirical methods. Perhaps the 

 theories may be improved. It is to be 

 hoped that the treatment of the differential 

 equations may be made more general and 

 complete. Efforts have been made in this 

 direction by Newcomb and others, and 

 especially by Gylden, but so far without 

 much practical result. 



The problem of three bodies was en- 

 countered by the mathematicians who fol- 

 lowed Newton, and many efforts were 

 made to solve it. These efforts continue, 

 although the complete investigations of 

 Lagrange appear to put the matter at 

 rest. The only solutions found are of very 

 special character. Laplace used one of 

 these solutions to ridicule the doctrine of 

 final causes. It was the custom to teach 

 that the moon was made to give us light 

 at night. Laplace showed by one of the 

 special solutions that the actual conditions 

 might be improved, and that we might 

 have a full moon all the time. But his 

 argument failed, since such a system is 

 unstable and cannot exist in nature. But 

 some of the efforts to obtain partial solu- 

 tions have been more fruitful, and G. W. 

 Hill has obtained elegant and useful re- 

 sults. These methods depend on assumed 

 conditions that do not exist in nature, but 

 are approximately true. The problem of 

 two bodies is a case of this kind, and the 



