A Treatise on the Theory of Screws

478 [431, THE THEORY OF SCREWS. 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 ß 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 2aß P~ a? + ßP If either a or ß 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 -- dß, where ß is zero and where d is infinite. In this case 2«£ a2 + ß'2 ''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 = cos0 and /3 = sin$; the pitch then assumes the very simple form sin 20. We thus see that the pitch can never exceed unity. If the motor be a vector, then we have ß= + a, or 0= + 45°, and the pitch is simply + 1. It should be noticed that a rotation a about the line A, and a rotation ß about its conjugate polar B, constitute a motor of the same pitch as a rotation ß 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