277] EM AN ANTS AND PITCH INVARIANTS. 293 



and L 2 + M 2 -\- N*= 1, whence we have for the pitch the homogeneous ex 

 pression 



It appears from this that the three equations, 



57,0 = ; -GTf fl = ; Tf fl = 0, 



indicate that 6 must be one of a pencil of rays of zero pitch radiating from 

 a point. 



The equations Z = ; lf=0; N=Q, 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. 



