148 



V. OSCILLATIONS AND CYCLIC MOTIONS. 



Fig. 29. 



Thus the differential equation of the cycloidal motion is 



d?s a 



25") 5- 4- S = 0, 



dt 2 r 4a 



showing that the motion is harmonic with the period 2jr J/ - as 



found above. The cycloid is isochronous for all arcs, the circle only 

 for infinitely small arcs. The circle having the same curvature as 



the cycloid at its 

 vertex is less steep 

 than the cycloid 

 (Fig. 29) and there- 

 fore the time of 

 descent on the circle 

 is greater for larger 

 arcs, as shown in 

 22. The evolute or envelope of the normals of a cycloid is an 

 equal cycloid, hence the cycloidal pendulum may be realized. If two 

 material half - cycloids be constructed tangent at (Fig. 30), where 

 the string is attached, and the string be allowed to wind itself 

 against them, if its length is that of the half -cycloid, its end will 

 describe a cycloid. This pendulum was constructed by Huygens. 1 ) 



The length being 4 a agrees with the above. On account of the 

 motion on the cycloid being harmonic Thomson and Tait call har- 

 monic motions cycloidal. 



43. Damped Oscillations. Let us now consider a particle 

 under the influence of a force proportional to its displacement from 

 a certain point and directed toward the position of equilibrium, the 



motion being resisted by 

 a non- conservative force 

 proportional to the first 

 power of the velocity. 

 Calling the accelerations 

 produced by i~he positional 

 (conservative) force h 2 s, 

 and the motional (non- 

 conservative) force 



ds 



Fig. 30. 



the equation of motion is 



26) 



1) Huygens, Horologium oscillatorium , Paris 1673. 



