DYNAMICS OF A RIGID BODY. 89 



also A must be reciprocal to a, so that its position on 

 the cylindroid is known ( 28). Finally, as the inten- 

 sity f\ f is given, and as the three screws ?/, A, /u are all 

 known, the intensity A" becomes determined ( 17). 



84. Small Oscillations. We shall now suppose that a 

 rigid body which has freedom of the first order occu- 

 pies a position of stable equilibrium under the influence 

 of a system of forces, as restricted in 6. If the body 

 be displaced by a small twist about the screw a which 

 prescribes the freedom, and if it further receive a small 

 initial twist velocity about the same screw, the body 

 will continue for ever to perform small twist oscillations 

 about the screw a. We propose to determine the time 

 of one oscillation. 



The kinetic energy of the body, when animated by a 



twist velocity -- is Mu^i-^} ( 59). The potential 



energy due to the position attained by giving the body a 

 twist of amplitude a away from its position of equili- 

 brium, is Fv^a* (72). But the sum of the potential and 

 kinetic energies must be constant ( 6), whence 



- + Fvja'* = const. 

 \dt J 



Differentiating we have 



Integrating this equation we have 



/ Fv ' z 



a = A sin \-^-t + B cos 

 <jMu* 



where A and B are arbitrary constants. The time of 

 one oscillation is therefore 



