Oscillations of various Periods. 371 



to inquire under what conditions the theorems remain intact 

 when the impressed force is a harmonic function of the time. 



As regards the first theorem, the justification for the neglect 

 of F and V may be that they are non-existent, as in many 

 problems of ordinary hydrodynamics. In such cases the 

 motion is at any instant' of the same character as if it had 

 been generated impulsively from rest, and the moment of 

 inertia is a minimum. But even when F and Y are generally 

 sensible, their influence tends to diminish as the frequency of 

 alternation increases, and we approach at last a state of things 

 in which they may be neglected. From this point onwards 

 we may say that the moment of inertia is a minimum in the 

 unconstrained condition. Thus in a system of electrical con- 

 ductors subject to a rapidly periodic electromotive force, the 

 distribution of currents is ultimately independent of the resist- 

 ances, and the self-induction is a minimum in the absence of 

 constraint. 



In like manner, even when T and F are sensible, the motion 

 tends to be more and more determined by V, as the frequency 

 of the vibrations is imagined to diminish. An " equilibrium 

 theory " ultimately becomes applicable, and the " stiffness " 

 is a minimum when there are no constraints. 



The theorem in which F is mainly concerned stands in a 

 somewhat different position. If T and Y are both sensible, we 

 cannot find an extreme case, in respect of the frequency of the 

 vibration, which shall annul their influence. If, however, Y 

 vanish, we can make F paramount by taking the period suffi- 

 ciently long ; and if T vanish, we can attain the same object 

 by limiting ourselves to the case when the period is very 

 short. If T and Y both vanish, the theorem of minimum 

 resistance in the absence of constraint holds good for all 

 periods of vibration. In the application to a system of elec- 

 trical conductors which possess resistance and induction, but 

 no capacity for charge needing to be regarded, we find that 

 while (as already stated) the induction becomes paramount 

 when the vibrations are very rapid, on the other hand when 

 they are very slow the distribution is determined ultimately 

 by the resistances only. In the first case the self-induction, 

 and in the second the effective resistance, is a minimum in 

 the absence of constraints. 



We are now prepared to enter upon the consideration of 

 the problem which is the main subject of the present paper, 

 viz. the behaviour of systems in which F, and one or other of 

 the two remaining functions T and Y, are sensible, but with- 

 out the restriction to very rapid or to very slow motions by 



2D2 



