108, 109] Rotating Cylinders 107 



The different intersections between the parabola and the other curves 

 will represent the different points of bifurcation. And, just as Poincare 

 has shewn to be the case in the three-dimensional problem, so we see here 

 that there is one point of bifurcation of each of the orders n = 3, 4, 5, . . . , and 

 that they occur in this order. 



The elliptical series accordingly loses its stability at the point of bifurca- 

 tion n = 3. The position of this point is obtained by solving equation (306) 

 with n put equal to 3, and the solution is readily found to be a = -J. 



From equations (303) and (304) we find that at this point of bifurcation 

 ft> 2 =f7rp, and Gf 2 =f' The configuration at the point of bifurcation is ac- 

 cordingly the elliptic cylinder 



fr = a'+*(P + '7 i ) ........................ (307), 



or, in Cartesian Coordinates, 



?/ 2 =5a 2 ........................... (308). 



109. Near the point of bifurcation, the configuration of the new linear 

 series will be determined by an equation of the form 



f7 7 = a 2 +|(^ + ^)+a 3 (r + 7 7 3 ) + a 1 (? + 7 7 ) ......... (309), 



and this is at once seen to be analogous to the pear-shaped series in three- 

 dimensions. The problem before us is to extend this series as far as possible 

 in the hope, which will be found to be fully justified by the event, that the 

 series will be found to be closely analogous to the three-dimensional series. 



Let us assume for the general configuration an expansion of the form 



=i 



where 6 is a parameter which vanishes at the point of bifurcation and con- 

 tinually increases as we pass along the series. The corresponding value 

 of to may be supposed given by 



^ + g 6 6 + .................. (311), 



it being immediately found that terms in 6, <9 3 , ... are unnecessary. It has 

 already been seen that S , which is the value of 1 &> 2 /27r/9 at the point of 

 bifurcation, is equal to f . 



Let us suppose that equation (310), solved explicitly for f, has the 

 solution 



f-(l-^) <&+& + &*+ 6*+-) ......... (312), 



where f is the already known value of f when 6 = 0, and f g is a general 

 series of ascending and descending powers of TJ, say 



?s = s + Srf + s 2 ij z + . . . + S-i*/" 1 -f 5_ 2 ^~ 2 + ............ (313). 



