24 



General Dynamical Principles 



[CH. n 



although competent to determine when stability ceases, cannot in general 

 determine what will happen after stability has ceased. In the same way a 

 general discussion will readily shew that a stick standing vertically on its 

 point is in unstable equilibrium, but it cannot determine in which precise 

 direction the stick will fall. 



22. In his classical paper* in which the theory of linear series and 

 points of bifurcation was first developed, Poincare used analytical methods 

 to obtain results identical with those just found. 



Consider a configuration in which the variable parameter has the value /x. 

 The potential .energy W will be of the form 



W=f(0 lt 2) ... O n) fl) 



and the configurations of equilibrium are given by the equations 



~r/\f(6\, 2 , ... 0n, A&) = 0, etc (8). 



As in 17, let ,, 2 , ... be a configuration of equilibrium corresponding 

 to this given value //, of the parameter, so that at the point lt 2 , ... n , /JL, 



At any adjacent point j + 80,, 2 + S0 2 , ... p + S/n, the value of W may 

 be expressed in the form 



The condition that this new configuration shall be one of equilibrium is, 

 from equations (8), 



and similar equations. Writing TT 12 for d^W/dO^Bz and so on, the solution of 

 these equations is 



W ut W u , 7W^ ~'\...\ A 



where A is given by 



W u ,W 12 ,...W l 



w nt w u , . w 



.(13), 



and so is the Hessian of W with respect to the variables lt 0. 2 , ... n . 



* I.e. ante. 



