422 THE THEORY OF SCREWS. 



Introducing the value just obtained for T, 



[386- 



There must be one screw on the cylindroid, for which 



This screw will have the accelerations ^ and 2 , both zero, and thus we have 

 the following theorem : 



If a rigid body has two degrees of freedom, then, among the screws about 

 which it is at liberty to twist, there is one, and in general only one, which has 

 the property of a permanent screw. 



The existence of a single permanent screw in the case of freedom of the 

 second order seems a noteworthy point. The analogy here ceases between 

 the permanent screws and the principal screws of inertia. Of the latter 

 there are two on the cylindroid ( 84). 



387. Geometrical Investigation. 



Let N (fig. 43) be the critical point on the circle which corresponds to 

 the permanent screw ( 50). Let P be a screw 0, the twist velocity about 



Oi 



Fig. 43. 



which is 0. Let u e be a linear parameter appropriate to the screw 6, such 

 that Mu g 2 6 2 is the kinetic energy. 



Let O x and 2 be the two screws of reference on the cylindroid and for 

 convenience let the chord 00 Z be unity. Let the point Q correspond to 

 another screw &amp;lt;f&amp;gt;, then from 57 



Ptolemy s theorem gives 



PQ6(f&amp;gt; = 2 &amp;lt;l $102- 



Now let &amp;lt; be the adjacent screw about which the body is twisting in a time 

 8t after it was twisting about 6. 



