January 22, 1903] 



NATURE 



283 



the derivatives of the longitudes and latitudes with respect to 

 the time, but it leads to long and rather uncertain computations. 

 Moreover, it employs more data than are necessary, and thus 

 is a departure from the mathematical theory of the problem. 

 This method is ingenious, and by means of the derivatives it 

 gives an interesting rule forjudging of the distance of a comet 

 from the earth by the curvature of its apparent path, but a trial 

 shows that the method of Olbers is much shorter. Good pre- 

 liminary orbits can now be computed for comets and planets 

 without much labour. This, however, is only a beginning of 

 the work of determining their actual motions. The planels act 

 on each other and on the comets, and it is necessary to compute 

 the result of these forces. Here again the conditions of our 

 solar system furnish peculiar advantages. The great mass of the 

 sun exerts such a superior force that the attractions of the 

 planets are relatively small, so that the first orbits, computed by 

 neglecting this interaction, are nearly correct. But the inter- 

 actions of planels become important with the lapse of time, and 

 the labour of computing these perturbations is very great. This 

 work has been done repeatedly, and we now have good 

 numerical values of the theories of the principal planets, from 

 which tables can be made. Practically, therefore, this question 

 appears to be well toward a final solution. But the whole story 

 lias nut been told. 



The planets, on account of their relative distances being great 

 and because their figures are nearly spherical, can be considered 

 as material particles, and then the equations of motion are 

 readily formed. In the case of n material particles acting on 

 each other by the Newtonian law, and free from external action, 

 we shall have yi differential equations of motion, and 6« in- 

 tegrations are necessary for the complete solution. Of these 

 only ten can be made, so that in the case of only three bodies 

 there remain eight integrations that cannot be found. The 

 early investigators soon obtained this result, and it is clearly 

 stated by Lagrange and Laplace. The astronomer, therefore, is 

 forced to have recourse to approximate methods. He begins 

 with the problem of two bodies, the sun and a planet, and neg- 

 lects the actions of the other planets. In this problem of two 

 bodies, the motions take place in a plane, and the integrations 

 can all be made. Two constants are needed to fix the position 

 of the plane of motion, and the four other constants pertaining 

 to the equations in this plane are easily found. This solution is 

 the starting point for finding the orbits of all the planets and 

 comets. The mass of the sun is so overpowering that the 

 solution of the problem of two bodies gives a good idea of the 

 real orbits. Then the theory of the variation of the elements 

 is introduced, an idea completely worked out into a practical 

 form by Lagrange. The elements of the orbits are supposed 

 to be continually changed by the attractions cf the other planets. 

 By means of this theory, and the mathematical machinery given 

 by Lagrange, which can be applied to a great variety of ques- 

 tions, the observations 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 solution. But this solution depends 

 on the use of series, which undergo integrations that may in- 

 troduce small divisors. An examination of these series by 

 Hansen, Poincare and others indicates that some of them are 

 not convergent. Hence the conclusions formerly drawn about 

 the stability of our solar system are not trustworthy and must 

 be held in abeyance. But looking at the construction of our 

 system, and considering the manner in which it was probably 

 evolved, it appears to be stable. However, ^he mathematical 

 proof is wanting. In finding the general integrals of the motions 

 of 11 bodies, the assumption 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 progressive motion of the system of bodies. They show 

 that the centre of gravity of the system moves in a right line 

 and with uniform velocity. Accurate observations of the stars 

 now expend 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 integrals 

 belong to the theory of areas, and Laplace has drawn from them 

 his theory of the invariable plane of the system. The remain- 



NO. 1 734, VOL. 67] 



ing integral gives the equation of living force. The question of 

 relative motion remains, and is the problem of theoretical 

 astronomy. This has given rise to many beautiful mathematical 

 investigations and developments into series. But the modern 

 researches have shown that we are not sure of our theoretical 

 results obtained in this way, and we are thrown back on em- 

 pirical 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 h) 

 Gylden, but so far without much practical result. 



The problem of three bodies was encountered by the mathe- 

 maticians who followed Newton, and many efforts were made 

 to solve it. These efforts continue, although the com- 

 plete 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 solutions have 

 been more fruitful, and G. W. Hill has obtained elegant and 

 useful results. 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 partial 

 solutions may illustrate, but will not overcome, the fundamental 

 difficulty. 



The arrangement of our solar system is such that the distances 

 of the planets from one another are very great with respect to 

 their dimensions, and this facilitates very much the determin- 

 ation of their motions. Should two bodies approach very near 

 each other, the disturbing force might become great, even in the 

 case of small masses. In the case of comets, this condition 

 happens in nature, and the comet may become a satellite of a 

 planet and the sun a disturbing body. In this way, it is probable 

 that comets and meteoric streams have been introduced into our 

 solar system. We have here an interesting set of problems. 

 This question is sometimes treated as one of statics, but since 

 the bodies are in motion it belongs to dynamics. Further study 

 may throw light on some relations between the asteroids and 

 the periodical comets. 



The great question of astronomy is the complete and rigorous 

 test of the Newtonian law of gravitation. This law has repre- 

 sented observations so well daring a century and a half that it 

 is a general belief that the law will prove true for all time and 

 that it will be found to govern the motions of the stars as well 

 as those of our solar system. The proof is cumulative and 

 strong for this generalitj. It will be a wonderful result if this 

 law is found rigorously true for all time and throughout the 

 universe. Time is sure to bring severe tests to all theories. We 

 know that the law of gravitation is modified in the motions of 

 the matter that forms the tails of comets. There is an anomaly 

 in the theory of Mercury which the law does not explain, and 

 the motion of our moon is not yet represented by theory. The 

 lunar theory is very complicated and difficult, but it does not 

 seem probable that the defect in Hansen's theory will be found 

 by recomputing the periodical coefficients, that have been already 

 computed by many mathematicians and astronomers, and with 

 good agreement by Hansen and Delaunay, by very different 

 methods. Hansen was a computer of great skill, but he may 

 have forced an agreement with observations, from 1750 to 1S50, 

 by using a coefficient of long period with an erroneous value. 

 No doubt the error of this theory will be discovered. Btck of 

 all theories, however, remains the difficulty of solving the 

 equations of motion so that the result can be applied with cer- 

 tainty over long periods of time. Until this is done, we shall 

 not be able to subject our law to a crucial test. 



The constants that enter the theories of the planets and 

 moon must be found from observations. In order to compare 

 observations made at distant epochs, the motions of the planes 

 of reference must be known with accuracy, and also the motion 

 of our solar system in spare. As the stars are our points of 

 reference, their positions and their proper motions mu^t be studied 

 with great care. This department of astronomy was brought to 

 a high degree of order by the genius of Bessel, whose work 

 forms an epoch in modern astronomy. The recent progress 

 made in determining the positions of the stars in all parts of the 

 heavens will be a great help to the investigations of the future. 



