105-107] Rotating Cylinders 105 



corresponding to a distortion of the circular section such that a n alone occurs 

 in equation (293). The different points of bifurcation correspond to the 

 different integral values of w. 



107. The value n = 1 may be rejected at once, since the corresponding 

 displacement is merely a rigid-body displacement of the cylinder when at 

 rest. Thus the first real point of bifurcation is given by n = 2. At this 

 point 



and here the series of circular configurations loses its stability. The branch 

 series has for its equation initially, 



r 2 = a 2 + 2a 2 r 2 cos 6 (301), 



so is of elliptical cross-section. 



When a 2 is small, the value of 6 2 has been seen to be a 2 . But when the 

 boundary is determined by equation (301), the values of the 6's are easily 

 determined, whether a 2 is small or not. Equation (293) reduces to 



of which the solution is of the form 



where a is a root of 



a = a 2 (l + a 2 ) (302). 



Thus the general value of 6 2 is Ja and all the other 6's vanish. The 

 equations of equilibrium (299) can accordingly be satisfied by a surface of 

 boundary (301) for all values of a. 2 . Thus the branch series through the 

 point of bifurcation just found is a series such that a 2 varies from to oo in 

 equation (301). The configurations form a series of elliptic cylinders, which 

 are obviously the two-dimensional analogue of the Jacobian ellipsoids. 



The conditions of equilibrium (299) reduce to the single equation 



, (303) 



which on combination with (302) gives 



<w 2 



Thus as we pass along this elliptic series, o> 2 decreases from TT/J to 0. The 

 angular momentum is however found to increase, so that the series is initially 

 stable. 



