34 Proceedings of the Royal Irish Academy. 



system is started in a slightly diffeient state, its subsequent motion is con- 

 fined to those slightly different states for what the total energy differs from 

 the maximum or minimum value by the same amount as at starting. This 

 general theorem, like the energy-test of the stability of equilibrium, applies 

 to cases in which the number of coordinates is infinite ; but in steady motion, 

 although it thus appears that the periods of the fundamental free disturbances 

 are real if the total energy is a maximum or a minimum, yet, in contra- 

 distinction to equilibrium, the converse is not true ; the free periods may be 

 real and yet the energy not a maximum or minimum. As far as I am aware, no 

 theorem imposing any limitation on the amount of deviation from the steady 

 state which is possible when nothing more is known than that the free 

 periods are real, has been established in such a form as to hold when the 

 number of coordinates is infinite; and accordingly I think it has not been 

 established that in such a case reality of the free periods constitutes a suffi- 

 cient condition of stability, and this whether there is, or is not, a potential 

 energy-function. 



Even when the number of coordinates is small — as small as two — the 

 system may be such that a large deviation from the steady state may result 

 from a small initial disturbance, although the periods are real and very 

 unequal. (It is, of course, known that this may happen if two periods are 

 nearly equal.) 



Suppose, for instance, a system in which the most general deviation from 

 the steady motion is expressed by the equations 



_' X - a cos {^t -^ a) ^l cos {qt + /3), 

 ' y = a sin (j:>/^ + a) + Jchpq''^ sin {qt + j3), 

 wherein x, y are coordinates which vanish in the steady motion, and 

 a, h, a, j3 are arbitrary constants, but h is a definite constant, nearly equal 

 to unity. Suppose the particular solution taken is 

 X = a (cos 2)t - cos qt), 



; ( sin 'pi - — sin qt \ 



giving X = a{-x>miiit + qsixiqt), 



y = aj) {cos'pt - k cos qt). 

 The system starts in a position which occurs in the steady motion and 

 with a disturbed velocity in tlie y coordinate alone ; and we see that if p and 

 q are such that we can have simultaneously cospt = + 1, cos qt = - 1, the 

 disturbance in the velocity in this direction exceeds its initial value in the 

 ratio (1 + h)l{l - k), which may be exceedingly great. 



It is easy to formulate, and in a variety of ways, kinetic and potential 

 energy-functions which lead to the above solutions. 



