74 Prof. J. D. Everett's Elementary Investigations 



position will then be greater in the outward than in the return 

 movement; the work done by the force in the outward move- 

 ment will therefore not be equal to the work of opposite sign 

 which it does in the return movement ; and this difference of 

 work will produce a steady gain or steady loss of energy from 

 one passage through the mean position to the next, which is 

 contrary to the supposition of a permanent state having been 

 attained. 



3. The external force then is a maximum in the extreme 

 position; and as this maximum force may either act outwards 

 or inwards, two opposite cases are possible. Either the ex- 

 ternal force urges the body outward during the whole of the 

 outward movement, and by continuing to act outward opposes 

 the motion during the whole of the return movement; or else 

 it opposes the whole of the outward movement and assists the 

 whole of the return. In this latter case the external force 

 always urges the body towards the mean position, and this is 

 precisely what the internal forces (on which free vibration 

 depends) are doing. Moreover, since the displacement, the 

 internal force, and the external force are all three of them 

 simple-harmonic functions of the time, with the same period, 

 and they attain their maxima simultaneously, the external force 

 has a constant ratio to the internal force. 



4. The acceleration due to the internal force is fi^ ; the 

 acceleration due to the external force must therefore have the 

 value fas, where fa is a constant. The actual acceleration 

 will be (fa + fa)s 9 directed towards the mean position. 



This reasoning also applies to the other alternative, except 

 that /jl 2 must be regarded as negative, because the acceleration 

 due to the external force will always be/rom the mean posi- 

 tion, and therefore opposite in direction to that due to the 

 internal force. In this case fa must be less in absolute mag- 

 nitude than fa ; for otherwise the actual acceleration would be 

 from the mean position — a supposition inconsistent with vibra- 

 tory movement. 



In all the foregoing reasoning, the displacement s may 

 either be a linear distance, or an angular distance, or any 

 other quantity which serves to measure the displacement of a 

 body with one degree of freedom. 



5. Denoting the actual period of vibration by T, and the 

 period of free vibration by T 1? we have 



T=2tt/ n /(> 1 + / * 2 ), T^tt/v^i- 



If the forced vibrations are quicker than the free vibrations, 

 T is less than T 1? and fa + fa must be greater than fa, hence 

 the external force always acts inward; but if the forced vibra- 



