156] PLANE REPRESENTATION OF DYNAMICAL PROBLEMS. 143 



155. Determination of the Wrench evoked by a Twist. 



The theorem just enunciated provides a simple means of discovering the 

 wrench which would be evoked by a small twist which removes the body 

 from a position of equilibrium. 



Let A (Fig. 33) be the given screw; join AO&quot;, and find H; then the 

 required screw A must be reciprocal to H, and is, accordingly, found by 

 drawing the chord HA through 0. 



Fig. 33. 



The axis 00&quot; is of course the homographic axis of 151. We need not 

 here repeat the demonstration of 141, which will apply, mutatis mutandis, 

 to the present problem. We see that the ratio of the intensity of the 

 wrench to the amplitude of the twist is proportional to 



HO 

 HO&quot; 



The other constructions of a like character can also be applied to this case. 



156. Harmonic Screws. 



If after displacement the rigid body be released, and small oscillations 

 result, the present geometrical method permits us to study the resulting 

 movements. 



It has been shown ( 130) that there are two special screws on the surface, 

 each of which possesses the property of being a harmonic screw. If a body 

 be displaced from rest by a small twist about a harmonic screw, and if it 

 also receive any small initial twist velocity about the same screw, then the 

 body will continue for ever to perform harmonic twist oscillations about the 

 same screw. 



The two harmonic screws are X and T, where the circle is intersected 

 by the axis passing through the pole of the axis of inertia , and the pole 

 of the axis of potential 0&quot; (Fig. 34). 



