A Treatise on the Theory of Screws

277] EMANANTS AND PITCH INVARIANTS. 293 and 7? + M2 + N2=l, whence we have for the pitch the homogeneous ex- pression _ 2L^vg + 2 Mtsr^g + Pe ~ L2+ M2 + N2 It appears from this that the three equations, = 0 ; = 0 J = 0, indicate that 0 must be one of a pencil of rays of zero pitch radiating from a point. The equations L = 0 ; Jf = 0; N = 0, define a screw of indeterminate pitch. Why the screws in the plane at infinity (§ 46) should in general present themselves with indeterminate pitch is a point which requires some ex- planation. The twist about such a screw, as around any other, consists, of course, of a rotation and a translation. If, however, the finite parts of the body are only to be moved through a finite distance, the amplitude of the twist must be infinitely small, for a finite rotation around an axis at infinity would, of course, imply an infinitely great displacement of parts of the body which were at finite distances. The amplitude of the rotation is therefore infinitely small, so that, if the pitch is finite, the displacement parallel to the axis of the screw is infinitely small also. It thus appears that the effect of a small twist about a screw of any finite pitch at infinity is to give the finite parts of the body two displacements, one of which is infinitely insig- nificant as regards the other. We can therefore overlook the displacement due to the pitch, and consequently the pitch of the screw unless infinite is immaterial; in other words, in so far as the screw is the subject of our investigation, its pitch is indeterminate. In like manner we can prove that a screw in the plane at infinity, when regarded as the seat of a wrench, must, when finite forces are considered, be regarded as possessing an indeterminate pitch. For, let the force apper- taining to the wrench be of finite magnitude, then its effect on bodies at finite distances would involve a couple of infinite moment. It therefore follows that the force on the screw at infinity must be infinitely small if the effects of the wrench are finite. The moment of the couple on the screw of finite pitch is therefore infinitely small, nor is its magnitude increased by importation from infinity; therefore, at finite distances, the effect of the couple part of the wrench may be neglected in comparison with that of the force part of the wrench. But the pitch of the screw is only involved so far as the couple is concerned; and hence whatever be the pitch of the screw lying in the plane at infinity, its effect is inoperative so far as finite operations are concerned.