478 THE THEORY OF SCREWS. [431, 



431. The Pitch of a Motor. 



Any small displacement of a rigid system in the content can be produced 

 by a rotation (see 417) a about one line followed by a rotation ft about its 

 conjugate polar with respect to the infinite quadric, the amplitudes of both 

 rotations being small quantities. The two movements taken together 

 constitute the motor. It will be necessary to set forth the conception in 

 the theory of the motor, which is the homologue of the conception of pitch 

 in the Theory of Screws in ordinary space. The pitch can most conveniently 

 be expressed by the function 



2ct/3 

 P ~a? + 0*- 



If either a. or ft vanish, then the pitch becomes zero. The motor then 

 degenerates to a pure rotation about one or other of the two conjugate 

 polars. This, of course, agrees with the ordinary conception of the pitch, 

 which is zero whenever the general screw motion of the rigid body degrades 

 to a pure rotation. 



In ordinary space we have 



pa. = dft, 



where ft is zero and where d is infinite. In this case 



&quot;a~ d 

 i.e. the pitch is proportional to the function now under consideration. 



No generality will be sacrificed by the use of a single symbol to express 

 the pitch. We may make a = cos# and /3 = sin#; the pitch then assumes 

 the very simple form sin 26. We thus see that the pitch can never exceed 

 unity. 



If the motor be a vector, then we have ft = a, or 6 = 45, and the 

 pitch is simply + 1. 



It should be noticed that a rotation a about the line A, and a rotation 

 ft about its conjugate polar B, constitute a motor of the same pitch as a 

 rotation ft about A and a about B. 



432. Property of Right and Left Vectors. 



To take the next step it will be necessary to discuss some of the relations 

 between right and left vectors. A right vector will displace any point P 

 in a certain direction PA ; a left vector will displace the same point in the 

 direction PB. It will, of course, usually happen that the directions PA 

 and PB are not identical. It is, however, necessary for us to observe 





