1 66 DYNAMICS OF A RIGID BODY. 



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 of Professor Sylvester (p. 743) are 

 presented in the present system 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 inter- 

 secting 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 re- 

 ciprocal screw, then all the screws of the cylindroids 

 form a screw complex 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 is 

 due to Professor Sylvester. 



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. 



154. Impulsive Screws and Instantaneous Screws. We 

 can determine the instantaneous screw corresponding to 

 a given impulsive screw in the case of freedom of the 

 fifth order by geometrical considerations. Let A, as 

 before, represent the screw reciprocal to the freedom, and 

 let p be the instantaneous screw which would correspond 

 to A as an impulsive screw, if the body were perfectly 

 free ; let r\ be the screw on which the body receives an 



