115, lie] Rotating Cylinders 115 



When e = ^, the intercepts are represented by the line A'E' ; the centre of 

 gravity still being on the line 00'. Thus there is a slight elongation at one 

 end of the figure and a corresponding contraction at the other end. In fig. 19 

 on the next page the chain curve shews the complete figure for e = ^, the 

 curve being of course calculated from the complete equation (338), while the 

 continuous curve represents the undisturbed elliptic cylinder e = 0. 



The intercepts of the figure e = J are represented by the line CD in fig. 18, 

 and the complete figure is shewn in fig. 20. We are still within the limits 

 within which equation (340) gives a good approximation. 



For the value e=\, the approximate intercepts are represented by the 

 line EF with a new double intercept at P. For values of e greater than 

 unity, there are four intersections of the surface with its axis, so that the 

 surface consists of two detached parts. At e = 1 this detachment is just be- 

 ginning ; there are two parts represented by EP, PF, but these are still in 

 contact at P. The curve calculated from equation (340) for the case of e = 1 

 is shewn in fig. 21. For values of < other than zero the convergence is con- 

 siderably better than for </> = 0, and this circumstance enables us to determine 

 the greater part of this curve with better accuracy than the points E, F in 

 fig. 18*. It appears that the curve has not yet quite divided, but it is obvious 

 that it is just on the point of doing so. 



Finally fig. 22 shews two ellipses which, with an approximation similar to 

 that used in 60 65, may be regarded as figures of equilibrium in rotation 

 about one another. The axes of the greater are in the ratio 2 : 1 which 

 corresponds to a rotation 



455 .............................. (341). 



A glance will suggest the probability that this figure gives a good repre- 

 sentation of the stage succeeding that shewn in fig. 21. If so the value (341) 

 ought to represent the value of G> 2 /27r/o given by equation (339) when e is just 

 greater than 1. The series is not convergent enough for us to determine this 

 limit from equation (339) directly, but it is clear that the value (341) is a per- 

 fectly possible value. 



Thus we may with fair confidence assert that the two-dimensional series 

 ends by fission into two detached masses, and in view of the close parallelism 

 which we have discovered between the two-dimensional and the three- 

 dimensional problems, it seems highly probable that the three-dimensional 

 series also will end by a similar fission into detached masses. 



* For greater detail, see Phil. Trans. 200 A, p. 100. 



82 



