114 Pear-shaped Configurations of Equilibrium [CH. v 



In fig. 18, values of r are represented by abscissae and values of e by 

 ordinates, so that <E> (r, e) is a function of position in the plane, and the 

 equation 4> (r, e) = will be that of a curve in this plane. The values of r 

 and e for which equation (340) has been seen to give a tolerable approxima- 

 tion are those within the area, shaded in the figure, which is bounded by the 

 curve e = ' 1, r = l/e. 





O' 



\ 



\ 



Fig. 18. 



The thick curves in this figure represent the locus <I> (r, e} = calculated 

 from the terms actually written down in equation (340). So long as we do 

 not pass far beyond the shaded area, this curve will give a fair representation 

 of the position of the true curve <l> (r, e) = which would be obtained by the 

 inclusion of all terms in the series of equation (340). The most important 

 points on this curve are those at which de/dr = 0. These are the points 

 P, P' in the figure, and they may, with sufficient accuracy for our present 

 purpose, be taken to be r = 2, e = 1 and r 2, e = 1. 



116. The diagram given in fig. 18 enables us to watch the changes in the 

 lengths of the intercepts on the principal axis of the rotating cylinder as e 

 increases i.e. as we pass along the linear series of pear-shaped figures. 



When e=0, the intercepts are represented by the Hne AB, being the 

 centre of gravity. The complete figure is of course the elliptic cylinder whose 

 equation is (308), and OA, OB are each equal to the semi-major axis -v/5. 



