482 SCIENCE PROGRESS 



as terms of the second order, is independent of the eccentricity 

 and inclination, Principia^ Prop. 66). The orbit is a periodic 

 one when referred to a plane rotating uniformly with the sun's 

 apparent rate of motion, the period being a synodic month 

 (interval from one new moon to the next). The " elliptic " and 

 " inclinational " inequalities are " free " oscillations about this 

 periodic orbit, the annual equation is a " forced " oscillation 

 having as period of the disturbance an anomalistic year. 



M. H. Poincare, in his Methodes Nouvelles de la Mecanique 

 Celeste has dealt with some other special cases. He takes two 

 of the three bodies as having infinitesimal masses and obtains 

 a class of solutions when these bodies move in circles about the 

 third body. From this he proceeds to the consideration of cases 

 when the masses of the two "planets" are small but no longer 

 infinitesimal. Periodic solutions are arranged in three classes 

 and though it would appear that the initial circumstance of the 

 motion may often not give rise to exactly periodic forms yet it 

 will frequently happen that these forms are approximations of 

 which the actual motion may be regarded as a " perturbation." 

 He finds that four classes of periodic solution exist, depending 

 respectively on the four arbitrary constants: (i) the period of 

 the infinitesimal body; (2) the constant of energy; (3) the time 

 of conjunction ; (4) the longitude of conjunction. 



In the work referred to Poincare also discusses general ques- 

 tions, such as the cases where the existence of periodic orbits 

 may be inferred, their method of appearance and disappearance 

 and general laws as to their periods of oscillation, etc. 



Sir George Darwin has taken the constant of relative energy 

 of the motion of the infinitesimal body as his single arbitrary 

 quantity. No general method being known for the determina- 

 tion of periodic orbits, he has made a numerical determination 

 of as many cases as possible and has traced the changes of 

 form that correspond to different values of the relative energy. 



The three bodies he calls the Sun (S), Jove (/) and the planet 

 or satellite, respectively, and takes Jove of unit mass moving in 

 a circle of radius unity round the sun of mass 10, all three bodies 

 lying in the same plane. The third body is taken of infinitesimal 

 mass (a material particle), so that no question of its action on 

 the others can arise. Thus when referred to moving axes, both 

 the Sun and Jove can be represented as fixed points in 

 diagrams. The ratio 10: i for the Sun in terms of Jove is taken. 



