86 Proceedings of the Royal Irish Academy. 



Aet. 7. The Assum])tion is valid, if Steady Motion eo:pon€nticdly Stable ; not if 



exponentially Unstable. 



On consideration, it appears that this assumption may be shown to be 

 correct, provided the free disturbances have stability of the ordinary expo- 

 nential character ; but that it would be incorrect if, for instance, any of them 

 were exponentially unstable or neutral ; this being so, the argument begs the 

 question at issue. For, if a system in an exponentially stable state, whether 

 of equilibrium or motion, be subjected to a simply harmonic disturbing force, 

 (or motion affecting a definite coordinate), of any definite period, the solution 

 in which the disturbance is simply harmonic and of the same period is known 

 to become asymptotically correct as the time increases indefinitely, whatever 

 may be the initial conditions (at least if the number of coordinates is finite). 

 When the disturbing force is exj^ressed as a Fourier integral, each element of 

 which is simply periodic in time, and the elementary periodic disturbances 

 which correspond to each in the fashion just described are combined by inte- 

 gration, it seems reasonable to infer that a similar statement would hold good 

 for the resulting integral disturbance. When the range of time through which 

 this resolution of the disturbing force is effected extends (say) from - 4 to + oo , 

 then, at any instant, t, this force has been in operation for a time t -v t^, even 

 though it may have been zero through a great portion of this interval, and 

 accordingly the solution obtained in this manner is, if the state be exponen- 

 tially stable, sufficiently accurate, provided t^t^i^ sufficiently great, whatever 

 may have been the disturbance (supposed finite) at the time - ^^o- But if the 

 disturbing force is zero from ^ = - ^^ to ?! = 0, then if the state is exponentially 

 stable, and tt^ is great enough, whatever finite disturbance may exist at the 

 time - ^0) it must be sensibly reduced to zero at ^ = ; so that in this mode 

 of procedure we do, indeed, obtain the solution in which there is no distur- 

 bance at the time zero. We have only to suppose t^ increased indefinitely to 

 obtain the case with which we have here to deal ; and hence it appears that 

 the value of i) determined from (2), (3), (8) does indeed satisfy (7). But this 

 argument fails, unless it is known that the state is exponentially stable. 



Art. 8. Mathematical Investigaiion of a simfple exam'fle ilhistrating Validity 



of this Objection. 



A simple instance of many which could be cited in which the analysis 

 is simple may serve to illustrate the argument, and especially to show 

 that the result need not hold for an unstable state; the elaboration of a 

 formal proof applicable to a case in which the number of independent 



