175-177] PLANETARY MOTION. 157 



4. A body, of mass km, describes an ellipse of eccentricity e and axis 

 major 2a under the action of a fixed gravitating body of mass m. Prove that, 

 if m is let go when the distance between the bodies is R, the eccentricity e of 

 the subsequent relative orbit is given by the equation 



e 2 -e 2 = 



5. Two gravitating particles of masses m, m are describing relatively to 

 each other elliptic orbits of eccentricity e and axis major 2a, their centre of 

 inertia being at rest. Prove that, if m is suddenly fixed when the particles 

 are at a distance R, the eccentricity e of the orbit subsequently described by 

 m is given by the equation 



l-e 2 \ /2 IN 

 - - -) = m - -- ). 

 l-e 2 J \fi a) 



177. General Problem of Planetary Motion. Consider 

 in the general case the motion of a system of particles which act 

 on each other according to the law of gravitation. The accelera 

 tion of any particle P is compounded of accelerations to each 

 of the others, and any one of the components thus arising is of 

 the form jm/r 2 , where m is the mass of one of the particles and r 

 is its distance from P. For such a system there would exist 

 seven first integrals of the equations of motion. In fact the 

 Principle of the Conservation of Linear Momentum would give 

 three such integrals, representing that the centre of inertia of 

 the system moves uniformly in a straight line ; the Principle 

 of the Conservation of Moment of Momentum would give three 

 integrals, representing that the moment of momentum of the 

 system about any one of the axes of reference is a constant 

 quantity ; the Principle of the Conservation of Energy would also 

 give an integral equation. 



But, even in the case of three particles, these integrals do not 

 suffice for a complete description of the motions of the system, and 

 up to the present time no other integral of the equations has been 

 obtained except for special circumstances of projection. 



Now the solar system affords an example of a system such as 

 that here considered, for the Sun and each of the planets are 

 approximately spherical, and it is consonant with results of obser 

 vation to assume that the component actions between the 

 particles of which, for purposes of Rational Mechanics, these 

 bodies are assumed to be made up, reduce to resultants in the 

 lines joining their centres of inertia. 



