102 



THE TIDAL PROBLEM. 



The derivative of (68) is 



dw K k{M — wy 



(70) 



It depends only upon w and is negative for all w>M. For w = M it is 

 infinite. For w less than, but near to, M it is positive; when M is large it 

 vanishes and becomes negative and again vanishes and becomes positive, 

 as w decreases from M to 0. It is positive for all w<0. From these facts 

 and equation (68) fig. 13 is drawn, the curves Ei, . . ., jE'^ belonging to four 

 values of E such that E^>Ei>E^>E^. 



Fio. 13. 



From equation (69) and fig. 13 the equi-energy curves in fig. 14 are 

 drawn, the corresponding curves being similarly lettered. There are two 

 points of particular interest, A and B. As the energy decreases, the energy 

 curves descend to a point at A. That is, on a section through A approxi- 

 mately parallel to the vE'-plane the point A is a minimum. With decreas- 

 ing E the energy curves separate at A and recede in opposite directions 

 nearly parallel to the liJ-axis. Hence a section by a plane through A approxi- 

 mately parallel to the ly^E^-plane has the point A as a maximum. Therefore 

 A. is a minimax point of the surface. 



The minimax point A, fig. 14, corresponds to the point A, fig. 13, at 

 which /(w)=0. Therefore, by (69), for this point 



v = 



K 



K+l 



W 



(71) 



