CASE X. 



113 



Those curves which are tangent to the axis are found by imposing the 

 condition (108) and 



!/i + 2/2 = e^ = (109) 



The first factors of (108) give no results of interest. Eliminating E between 

 the third factor and equation (109) we get 



P-(M-9)' = 



agreeing with (103), the conditions for a maximum or a minimum. If 

 equation (103) has no real roots then there is but one intersection of each 



m. 



curve with the D-axis to the right of D = — ^ When the equation has real 

 roots then for certain values of E there are three intersections to the right 



M 



m, 



As E approaches — oo the curves approach the line D = ^ 



between ?/ = and y= -\-l. 



We get the final equi-energy curves from 



e = '^^jy = '^^jyl + 



Vi 



(110) 



Fia. 16. 



They are given in fig. 15. The point A is one of the solutions of (103) 

 and is a minimax of E considered as a function of D and e. When this 

 point belongs to a positive value of E, as it does in the earth-moon system, 

 the curves for this and larger values of E are open on the right to infinity, 



because the curves are asymptotic to the lines e= ±-^ n^ . When 



E decreases until EM^=—k the curves close at + oo , and for smaller values 

 of E they are closed ovals until they vanish at a point B on the D-axis to the 

 right of A. This point corresponds to a true minimum of E considered as 

 a function of D and e. On the left of the e-axis the curves are shaped 



