144 THE THEORY OF SCREWS. [156- 



For, suppose the body receives a small initial displacement about X, this 

 will evoke a wrench on H, found by drawing XO&quot;Y and YOH ( 155). But the 



Fig. 34. 



effect of a wrench on H will be to produce twist velocity about a screw found 

 by drawing HOY and YO X, i.e. X itself ( 140). Hence the wrench evoked 

 can only make the body still continue to twist about X, and harmonic 

 vibration on X will be the result. Similar reasoning, of course, applies 

 to Y. 



157. Small Oscillations in general. 



The initial displacement, and the initial twist velocity of the body, can 

 always be decomposed into their respective components on X and Y. The 

 resulting small oscillations can thus be produced by compounding simple 

 harmonic twist oscillations about X and Y. 



If it should happen that and 0&quot; become coincident, then every screw 

 would be a harmonic screw. 



If and coincided, then every screw would be a principal screw of 

 inertia ( 86). 



If and 0&quot; coincided, then every screw would be a principal screw of 

 potential. 



158. Conclusion. 



The object proposed in this Chapter has now been completed. It has 

 been demonstrated that the representative circle affords a concise method 

 of exhibiting many problems in the Dynamics of a Rigid System with two 

 degrees of freedom, so long as the body remains near its initial position. 

 The geometrical interest of the enquiry is found mainly to depend on the 

 completely general nature of the constraints. If the constraints be specialized 

 to those with which mechanical problems have made us familiar, it will 



