102 DYNAMICS OF A RIGID BODY. 



95. Displacement of a Point. Let P be a point, and 

 let a, |3 be any two screws upon a cylindroid. If a body 

 to which P is attached receive a small twist about a, the 

 point P will be moved to P f . If the body receive a 

 small twist about )3, the point P would be moved to P '.. 

 Then whatever be the screw y on the cylindroid about 

 which the body be twisted, the point P will still be dis- 

 placed in the plane PP'P". 



For the twist about y can be resolved into two twists 

 about a and j3, and therefore every displacement of P 

 must be capable of being resolved along PP' and PP ' . 



Thus through every point P in space a plane 

 can be drawn to which the small movements of P, 

 arising from twists about the screws on a given cylin- 

 droid are confined. The simplest construction for this 

 plane is as follows : Draw through the point P two 

 planes, each containing one of the screws of zero pitch ; 

 the intersection of these planes is normal to the required 

 plane through P. 



The construction just given would fail if P lay upon 

 one of the screws of zero pitch. The movements of P 

 must then be limited, not to a plane, but to a line. The 

 line is found by drawing a normal to the plane passing 

 through P, and through the other screw of zero pitch. 



We thus have the following curious property of the 

 lines of zero pitch, viz., that a point in the rigid body on 

 the line of zero pitch will commence to move in the 

 same direction whatever be the screw on the cylindroid 

 about which the twist is imparted. 



This easily appears otherwise. Appropriate twists 

 about any two screws, a and ]3, can compound into a twist 

 about the screw of zero pitch A, but the twist about X 

 cannot disturb a point on X. Therefore a twist about )3 

 must be capable of moving a point originally on X back 

 to its position before it was disturbed by a. Therefore the 



