THE 

 LONDON, EDINBURGH, and DUBLIN 



PHILOSOPHICAL MAGAZINE 



AND 



JOUKNAL OF SCIENCE. 



[FIFTH SERIES.] 



FEBRUARY 1883. 



XII. Elementary Investigations relating to Forced Vibrations ; 

 with Applications to the Tides and to Controlled Pendulums. 

 By Prof. J. D. Everett, F.R.S., Queens College, Belfast*. 



1. " T ET there be a body whose free vibrations are simple- 

 J-J harmonic, the acceleration for a displacement s being 

 /i^s, and the period being therefore 2ttJ Wi« If this body be 

 acted on by an external force which is a simple-harmonic 

 function of the time, and which urges the body along the 

 same path which it would take if vibrating freely, the body 

 will ultimately be brought into a permanent state of simple- 

 harmonic vibration with the same period as the external force. 

 This is on the supposition (which is practically fulfilled in the 

 majority of actual cases) that, though friction can be neglected 

 in considering a single vibration, its accumulated effect in a 

 great number of vibrations is sufficient ultimately to destroy 

 any previously existing free vibration if the body were left to 

 itself. 



2. It is easy to show, by reductio ad absurdum, that in the 

 permanent state the maxima of the external force must coincide 

 with the maxima of displacement. Let us, for instance, try the 

 supposition that the force attains its maximum after the mean 

 position has been passed but before the extreme position has 

 been attained (and if this be true on one side, it must also, by 

 symmetry, be true on the other). The force in any given 



* CoHiiimnicated by the Author. 

 Phil. Mag. S. 5. Vol. 15. No. 92. Feb. 1883. G 



