THE PROBLEM OF THREE BODIES 483 



so that thereby all the phenomena of perturbation are greatly 

 exaggerated as compared with those of nature, and are thus more 

 clearly shown. 



He first obtains the Jacobian integral : 



(Cis a constant, Kdenotes the velocity of the particle relatively to 

 axes moving with angular velocity the same as that with which 

 Jove and the Sun move). The function /2 is the potential of the 

 system and so the equation V- = 2/2 — C is called the equation 

 of relative energy. Since in all real cases F- must be positive, 

 2/2 can never be less than C and accordingly the particle can 

 never cross the curve represented by 2/2 = C; if this curve has 

 a closed branch inside of which the particle is it must remain 

 inside, if outside it must remain outside. This agrees with 

 Hill's result in assigning superior and inferior limits for the 

 position of the moon. 



Since 2/2 also = v (r-- +"-) + [pr +-) where v is the sun's mass 



in terms of Jove (taken here as 10) and r and p are the distances 

 of the particle from the Sun and Jove respectively, we see that 

 when r and p are small the equation obtained by putting V = O ; 



2,V 2 



2/2 = C= — H — nearly, the curves of "zero velocity" are like 



the equipotential curves round two attracting centres of masses 

 2v and 2 and by putting various values for the constant C the}^ 

 may be drawn. Thus when C is large they are closed ovals 

 round Sun and Jove, the one round the former body being the 

 larger. As C diminishes the ovals swell and unite into a figure 

 of eight. 



When r and p are large the equation becomes approximately 



vr^ ■\- pr = C (a Cartesian oval). 



For large values of C the curve of zero velocity consists of two 

 closed branches round Sun and Jove respectively and a third 

 closed branch round both bodies. As C decreases the larger 

 oval shrinks and the inner ovals swell and unite into a figure of 

 eight which gradually takes a form like that of an hour-glass 

 with continually thickening neck. The outer and inner curves 

 meet at length, C continually diminishing, uniting by the smaller 

 bulb of the hour-glass figure (that around Jove) touching the 



