112 THE TIDAL PROBLEM. 



We shall now study the surface defined by the second equation of (100) 

 by considering the curves for various values of E. Consider first the curves 

 defined by 



y.=i-gi(M-^y (105) 



,.4(m-^)' (106) 



From 



i=t(«-'^')(*^-T) (^»^' 



we find the positions of the maxima and minima of y^ and we can then 

 easily construct its graph. 



We must now consider the curves defined by equation (106) for various 

 values of E, both positive and negative. They all pass through the point 



j^^ = 0, D = ^, and are tangent to the D-axis at this point. All the branches 



of the curves are asymptotic to the lines D = and y^^EWln, except in 

 the special case when E = 0. 



The sum y = yi + yi must then be considered. Whatever the value of 

 E, all of these curves are asymptotic to D = and y= — QC. All of the 



curves pass through the point D = ^, y=l, and their slope is zero at this 

 point. Their slope vanishes at the points defined by 



^^?^rM-^ir-^+^-?^l=0 (108) 



m. 



The solutions of this equation are D= ± oo , D=:-^, and the two roots of 



the last factor. When E is large the roots are small numerically, one being 

 positive and the other negative. With decreasing E the negative root 

 recedes to — oo which it attains at E = 0. It then becomes positive from 

 + (X) and unites with the other positive root when 



E= '''" 



4?/?' 

 The curve has a point of inflection for this value of E, for which ^ = -^y 



and for smaller values of E the slope is always negative for values of D 



greater than -^- 



