104 THE TIDAL PROBLEM. 



time the w belonging to the system is less than the abscissa of A, that is, 

 if the point is on the curve E^, it will always remain less than the abscissa 

 of A and will approach the point B as a limit; but if the w is greater than 

 the abscissa of A and less than M it will always remain between these two 

 values. The abscissa of every point on the oval part of E^ is in all possible 

 cases less than the abscissa of A, for the points on this branch of the curve 

 correspond to the points of the curve E^ of fig. 13 which are above the 

 u;-axis and to the left of the line w = M. This part of the curve is always 

 to the left of A, for the curves of fig. 13 are all derivable from any one of 

 them simply by vertical displacements. 



A simple case is that in which the two bodies are precisely alike in every 

 respect and have at any time similar motions. Then from the symmetry 

 of the problem the motions of the two bodies will always be the same. 

 Equations (55) and (56) become in this case 



The treatment of these equations is precisely like that of (23) and (24) 

 of section V if we replace m^ of that section by 2wi. 



