148 Mr. R. J. Pocock on the Action of a disturbing 



enough or distant enough for us to be able to neglect its 

 differential effect on H and J. Let SP = cr. 



Jacobins integral will, in general, no longer exist ; though 

 the equations will be of the same form provided we intro- 

 duce into the function H an additional term - ; where cr, 

 however, is an explicit function of the time t. a 



3. Confining ourselves as usual to two dimensional 

 motion, two cases may arise in which the integral will still 

 exist. 



(i.) There may be a number of disturbing bodies so related 

 that the time disappears from the function 12, e. a., the bodies 

 might be distributed in a circular belt with centre at the 

 origin such that the density at any point w r as independent of 

 the vectorial angle (though it might be a function of the 

 radius vector). The minor planets and Saturn's rings might 

 be very approximately represented by such a distribution. 



(ii.) The disturbing body, or bodies, might be at rest 

 relative to the rotating axes. Such will be the case when 

 bodies are placed at the points of zero force, and will hold 

 approximately of bodies oscillating near these points. We 

 have examples of such in the Jupiter groups, Hector, 

 Patroclus, &c. 



In either of these cases the curves of zero velocity will be 

 given by the equation 2Cl = C where C is an absolute 

 constant, and 



V r) r p a 



4. In the more general case, however, this integral will 

 not exist. Assuming, however, that the disturbing force is 

 not considerable, we may assume the integral Y" = 2Q, — C 

 where G is no longer constant, but a function of the 

 time t. 



At any given time t the instantaneous curves of zero 

 velocity will be given by '2Q = C. 



5. In the restricted problem of three bodies there exist 

 certain forms of stable motion, such for instance as that in 

 which P, if once an inferior planet, must always remain so. 

 In such a case the curves of zero velocity consist in two 

 closed ovals surrounding H and J. When a small dis- 

 turbing force acts these ovals will swell and shrink 

 periodically. Suppose for example that the disturbing force 

 is due to the attraction of a planet S superior to J. We are 



