Force in tlie Restricted Problem of Three Bodies. 14'.) 



not concerned with the actual path of S provided its 

 perihelion distance is greater than unity } and its differ- 

 ential effect on H and J is small. It may happen that the 

 two oval curves never quite coalesce during the period of the 

 disturbing force. In this case, although there is no fixed 

 curve of zero velocity, the particle once started as an 

 inferior planet must always remain so> The limits of the 

 initial values of C will not be the same as in the case when 

 8 is non-existent. In order to find these values we must 

 suppose that when S is at perihelion P is at the critical point 

 between H and J ; this is the most favourable case for 

 disturbing the stability of the motion. If C denotes the 

 critical value of for undisturbed motion and SP = o-, we 



have as the criterion for stability -<C — , 



G 



' 



c>c +-^ 



We may at once extend the theorem to any number 

 disturbing planets and we have 



6. If the disturbing body is inferior to J we have only to 

 treat J as the disturbing body and H and S as the two 

 principal bodies. Similarly we may deal with cases in 

 which P moves as a J satellite or as a planet superior to J 

 or in which interchange between two forms is possible. 

 We may introduce as many disturbing bodies as we please ; 

 it is only necessary to know in each case either the peri- 

 helion or the aphelion distance, and there is no diffi- 

 culty in determining numerical limits for the initial value 

 of C. 



7. In the restricted problem of three bodies there exist 

 certain points at which P can remain in relative equilibrium. 

 three on the lineHJ, and two at the apices of the equilateral 

 triangles described on HJ. Let L denote one of these 

 points. In the case of disturbed motion the actual critical 

 point 1/ will lie near L, and will not be fixed. For example, 

 if the disturbing force is due to a single superior planet S, 

 1/ will describe a small closed periodic curve round L. 



8. In the case of three bodies it is known that the motion 

 near such a point L on the line HJ is given by the 



