Ore — Stalility or Instability of Motions of a Perfect Liquid. 33 



disturbances, the reality of all the values of w is a satisfactory proof of 

 stability, only for the reason that it shows that, if there be a potential-energy 

 function it is essentially positive in any displacement (if taken as zero in 

 the equilibrium position); for the problem of finding the free periods is 

 analytically identical with that of transforming the coordinates, so that the 

 kinetic energy can be expressed as a linear function of the squares of the 

 velocities, and at the same time the potential energy as a linear function of 

 the squares of the coordinates, the terms involving products being thus made 

 to disappear ; and if all the periods are real, each coeiScient in the potential 

 energy-function is positive. Whether the potential energy is, or is not, a 

 minimum in the state of equilibrium can, of course, generally be decided 

 much more easily directly than by investigating the free periods. 



If we consider even a system ha^dng only a finite number of coordinates, 

 and which is slightly displaced from equilibrium, the argument for universal 

 stability which is derivable from the stability of the fundamental modes may 

 be very much weakened (i.e., the limits of stability may be very much 

 narrowed) by the non-existence of a potential-energy function. Take, for 

 example, two particles of equal mass oscillating in a straight line, and 

 subject to forces such that the most general small motion is given by the 



equations 



o: = A cos {yt + a) + B cos (qt + /3), 



y = A cos (jjt + a) + Bk cos {qt + j3.), 



wherein A, B, a, j3 are arbitrary, but h is a definite constant, nearly equal to 

 unity. Suppose that in the position of equilibrium a velocity is imparted to 

 the second particle only. The resulting motion is given by 



X = A\ sin pt - - sin qt ], 

 \ 9. J 



y ~ A\ sin jj^ - /j -s in qt 



and we see that if p, q are such that we can have simultaneously cos jj^ = + 1 

 cos qt = - 1, the maximum kinetic energy exceeds the initial in the ratio 

 {b + 2k ■\- k^)l{l - ky, which may be exceedingly great. If, however, the same 

 arbitrary constants A, B, a, (5 occur in the equations expressing the small 

 motions of a similar system having a potential-energy-function, k must have 

 the value - 1 ; and in consequence, if the system be started subject to the 

 same initial conditions, the kinetic energy can never exceed its initial value. 

 When the question is of the stability of a given state of motion, if the 

 state is one for which the sum of the kinetic and potential energies is a 

 minimum or maximum, then, whether steady or not, it is stable ; for if the 



K. I. A. PROC, VOL. XX^^I,, SECT, A. [5] 



