SURFACES OF DISCONTINUITY 605 



XIV. The Stability of Surfaces of Discontinuity 



38. Conditions for the Stability of a Dynamical System 



When the stabiHty of a dynamical system is being investi- 

 gated, the potential energy of the system is expressed as a 

 function of the coordinates of the system. If the system were 

 at rest in any configuration this function of the coordinates for 

 this configuration would give the whole energy of the system. 

 If this configuration is one of equilibrium then the partial 

 differential coefficients of the function with respect to different 

 coordinates are severally zero; for if /(xi, Xi, xs, . . .) represents 

 the function, Xi, Xi, Xz, ... being the coordinates, we know that 

 to the first order of magnitude f{x\^ Xi, xz, . . . ) must not vary in 

 value when xi, x^, Xz, ... receive small arbitrary increments 

 bxi, 8x2, dxz, . . . Thus 



9/ 9/ 9/ 



— 8x1 + — 8x2 + — 8xz+ . . . =0, 



dxi dX2 dxz 



and since 8x1, 8x2, 8xz, . . . are arbitrary, it follows that 



9/ 9/ df 



— = 0, r^ = 0, r" = 0, etc. 



dxi ' dx2 ' dxz 



We can express this simply by the condition 



8f{xi, X2, xz, . . . ) =0. 



Now the equilibrium may be stable, unstable or neutral. 

 If we wish to investigate the matter in more detail we must 

 consider the value of A/(a:i, X2, xz, . . .). This is equal to the 

 value of f(xi + 8x1, X2 + 8x2, xz + 8x3, . . . ) — f(xi, X2, xz, . . . ) 

 when higher powers of 8x1, 8x2, 8xz, etc. than the first are re- 

 tained in the expansion of f(xi + 8x1, X2 + 8x2, xz + 8xz, . . .). 

 In many cases it is sufficient to retain the second powers 

 and neglect those that are higher. For convenience we write 

 ^1, ^2, ^3, ... for 8x1, 8x2, 8xz, . . . Then by Taylor's theorem 



