A Treatise on the Theory of Screws

69-72] THE EQUILIBRIUM OF A RIGID BODY. 63 the ratios of the amplitudes of the twists about A,,... An and hence in the selection of a screw from the screw system of the nth order, we have n — 1 disposable quantities. 70. Constraints. An essential feature of a system of constraints consists in the number of independent quantities which are necessary to specify the position of the body when displaced in conformity with the requirements of the constraints. That number which cannot be less than one, nor greater than six, is the order of the freedom. To each of the six orders of freedom a certain type of screw system is appropriate. The study of the six types of screw system as here defined is a problem of kinematics, but the statical and kinematical properties of screws are so interwoven that we derive great advantages by not attempting to relegate the statics and kinematics to different chapters. We shall not require any detailed examination of the constraints. Every conceivable condition of constraints must have been included when the six screw systems have been discussed in their most general form. Nor does it come within our scope, except on rare occasions, to specialize the enunciation of any problem, further than by mentioning the order of the freedom permitted to the body. 71. Screw Reciprocal to a System. If a screw X be reciprocal to n independent screws, A,,... An, of a screw system of order n, then X is reciprocal to every other screw A which belongs to the same screw system. For, by the property of the screw system, it appears that twists of appropriate amplitudes about A>, ... An, would compound into a twist about A. It follows (§ 69) that wrenches on A,, ... An, of appropriate intensities (§ 30) compound into a wrench on A. Suppose these wrenches on Alt ... An, were applied to a body only free to twist about X, then since X is reciprocal to A,, ... An, the equilibrium of the body would be undisturbed. The resultant wrench on A must therefore be incapable of moving the body, therefore A and X must be reciprocal. 72. The Reciprocal Screw System. All the screws which are reciprocal to a screw system P of order n constitute a screw system Q of order 6 - n. This important theorem is thus proved:— Since only one condition is necessary for a pair of screws to be reciprocal, it follows, from the last section, that if a screw X be reciprocal to P it will fulfil n conditions. The screw X has, therefore, 5-n elements still dis- posable, and consequently (n < 5) an infinite number of screws Q can be