A Treatise on the Theory of Screws

338 THE THEORY OJF SCREWS. [312- 312. Freedom of the First or Second Order. If the rigid body have only a single degree of freedom, then the only- movements of which it is capable are those of twisting to and fro on a single screw a. If the impulsive wrench i] which acted upon the body happened to be reciprocal to a, then no movement would result. The forces would be destroyed by the reactions of the constraints. In general, of course, the impulsive screw r] will not be reciprocal to a. A twisting motion about a will therefore be the result. All that can be said of the instantaneous screw is that it can be no possible screw but a. In the next case the body has two degrees of freedom which, as usual, we consider to be of the most general type. It is required to obtain a con- struction for the instantaneous screw a about which a body will commence to twist in consequence of an impulsive wrench The peculiarity of the problem when the notion of constraint is introduced depends on the circumstance that, though the impulsive screw may be situated anywhere and be of any pitch, yet that as the body is restrained to only two degrees of freedom, it can only move by twisting about one of the screws on a certain cylindroid. We are, therefore, to search for the in- stantaneous screw on the cylindroid expressing the freedom. Let A be the given cylindroid. Let B be the system of screws of the fourth order reciprocal to that cylindroid. If the body had been free it would have been possible to determine, in the manner explained in the last section, the impulsive screw corresponding to each screw on the cylindroid A. Let us suppose that these impulsive screws are constructed. They will all lie on a cylindroid which we denote as P. In fact, if any two of such screws had been found, P would of course have been defined by drawing the cylindroid through those two screws. Let Q be the system of screws of the fourth order which is reciprocal to P. Select from Q the system of the third order Qi which is reciprocal to rj. We can then find one screw which is reciprocal to the system of the fifth order formed from A and ft. It is plain that V1 must belong to B, as this contains every screw reciprocal to A. Take also the one screw on the cylindroid A which is reciprocal to y, and find the one screw on the cylindroid P which is reciprocal to this screw on A. Since y, and % are all reciprocal to the system of the fourth order formed by Aj and Qlt it follows that i), rfo, and must all lie on the same cylindroid. We can therefore resolve the original wrench on r) into two com- ponent wrenches on and