A Treatise on the Theory of Screws

383] THE THEORY OF PERMANENT SCREWS. 419 will not depart, from the original instantaneous screw except by an accelera- tion. This must be on a screw which stands to the restraining screw in the relation of instantaneous to impulsive, but in the case supposed these two screws are reciprocal, therefore they cannot be so related, and therefore there is no acceleration. There is little to be said as to the restraining wrench when the freedom is of the first order. Of course, in this case, as every movement of the body can only be a twist about the screw which prescribes its freedom, the restraining wrench is provided by the reactions of the constraints. It is only where the body has liberty to abandon its original instantaneous screw that the theory of the restraining wrench becomes significant. 383. Two Degrees of Freedom. If a rigid body has two degrees of freedom, then it is free to twist about every screw on a certain cylindroid. If the body be set initially in motion by a twist velocity about some one screw on the surface, then, in general, it will not remain twisting about this screw. A movement will take place by which the instantaneous axis gradually comes into coincidence with other screws on the cylindroid. If we impose a suitable restraining wrench 7), then of course 3 can be maintained as the instantaneous screw; y is reciprocal to 3. It may be compounded with any reactions of the constraints of the system. Thus, given 3, there is an entire screw system of the fifth order, consisting of all possible screws reciprocal to 3, any one screw of which may be taken as the restrainer. Of this system there is one, but only one, which lies on the cylmdroid itself. There are many advantages in taking it as the restraining wrench, and it entails no sacrifice of generality ; we there- fore have the following statementTo each screw on the cylindroid, regarded as an instantaneous screw, will correspond one screw, also on the cylindroid, as a restraining screw. ______ The position of this restraining screw is at / / '• once indicated by the property that it must be / ot/ \ reciprocal to the instantaneous screw. It we / X \ employ the circular representation for the ; / Ji screws on the cylindroid (fig. 42), and if 0 be I / the pole of the axis of pitch, then it is known \ / that the extremities of any chord, such as 1R / drawn through 0, will correspond to two re- 1 ciprocal screws (§ 58). If therefore I be the instantaneous screw, R must be the restraining screw. If a body free to twist about all the screws on the cylindroid be set in motion by a twist velocity about I, it will be possible, by a suitable wrench 27—2 ■■■■