250 THE THEORY OF SCREWS. [230- 



It must always be possible to find a single screw X which is reciprocal 

 to the six screws P, Q, R, S, T, U. Suppose the rigid body were only free 

 to twist about X, then the six forces would not only collectively be in equi 

 librium, but severally would be unable to stir the body only free to twist 

 about X. 



In general a body able to twist about six screws (of any pitch) would 

 have perfect freedom ; but the body capable of rotating about each of the 

 six lines, P, Q, R, S, T, U, which are in involution, is not necessarily perfectly 

 free (Mobius). 



If a rigid body were perfectly free, then a wrench about any screw could 

 move the body; if the body be only free to rotate about the six lines in 

 involution, then a wrench about every screw (except X) can move it. 



The conjugate axes discussed by Sylvester are presented in the Theory of 

 Screws as follows : Draw any cylindroid which contains the reciprocal 

 screw X, then the two screws of zero pitch on this cylindroid are a pair of 

 conjugate axes. For a force on any transversal intersecting this pair of 

 screws is reciprocal to the cylindroid, and is therefore in involution with the 

 original system. 



Draw any two cylindroids, each containing the reciprocal screw, then all 

 the screws of the cylindroids form a screw system of the third order. 

 Therefore the two pairs of conjugate axes, being four screws of zero pitch, 

 must lie upon the same quadric. This theorem, due to Sylvester, is proved 

 by him in a different manner. 



The cylindroid also presents in a clear manner the solution of the 

 problem of finding two rotations which shall bring a body from one position 

 to any other given position. Find the twist which would effect the desired 

 change. Draw any cylindroid through the corresponding screw, then the 

 two screws of zero pitch on the cylindroid are a pair of axes that fulfil the 

 required conditions. If one of these axes were given the cylindroid would 

 be defined and the other axis would be determinate. 



231. Four Screws of a Five-system on every Quadric. 



On any single sheeted hyperboloid four screws of any given pitch p can 

 in general be determined which belong to any given system of the fifth 

 order. A pair of these screws lie on each kind of generator. 



Let X be the screw reciprocal to the system. Take any three generators 

 A, B, C of one system on the hyperboloid, and regarding them as screws of 

 pitch p draw the cylindroid XA and take on this A the second screw of 

 pitch p. Then the two screws of pitch p which can be drawn as transversals 

 across A, B, C, A are coincident with two generators of the hyperboloid 



