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investigators soon obtained this result, and it is clearly stated by La- 

 grange 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 neglects 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 jjosition 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 of 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 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 com- 

 plete solution. But this solution depends on the use of series, which 

 undergo integrations that may introduce small divisors. An examina- 

 tion 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 the mathematical proof is wanting. In finding the 

 general integrals of the motions of n 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 center 

 of gravity of the system moves in a right line, and with uniform 

 velocity. Accurate observations of the stars now extend over a cen- 

 tury 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 gen- 

 eral integrals 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 motion remains, and is the problem of theoretical astronomy. 



