L10 THE THEORY OF SCREWS. [121- 



121. 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 . If the body receive a small twist about /3, the point P 

 would be moved to P&quot;. Then whatever be the screw 7 on the cylindroid 

 about which the body be twisted, the point P will still be displaced in the 

 plane PP P&quot;. 



For the twist about 7 can be resolved into two twists about a and /3, and 

 therefore every displacement of P must be capable of being resolved along 

 PP and PP&quot;. 



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 

 cylindroid 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 due to M. Mannheim*, 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 X, but the 

 twist about X cannot disturb a point on X. Therefore a twist about ft must 

 be capable of moving a point originally on X back to its position before it 

 was disturbed by a. Therefore the twists about /3 and a. must move the 

 point in the same direction. 



122. Properties of the Pitch Conic. 



Since the pitch of a screw on a cylindroid is proportional to the inverse 

 square of the parallel diameter of the pitch conic ( 18), the asymptotes 

 must be parallel to the screws of zero pitch ; also since a pair of reciprocal 

 screws are parallel to a pair of conjugate diameters ( 40), it follows that 

 the two screws of zero pitch, and any pair of reciprocal screws, are parallel 

 to the rays of an harmonic pencil. If the pitch conic be an ellipse, there 



* Journal de I ecole Polytechnique, T. xx. cah. 43, pp. 57122 (1870). 



