A Treatise on the Theory of Screws

72 THE THEORY OF SCREWS. [81- the six products and remembering that a and f are reciprocal by hypothesis while a and p are reciprocal by the nature of the reactions of the constraints, we have Pi«ißi + ... + K«<A = 0. The symmetry of this equation shows that in this case y must be reciprocal to ß. Hence we have the following theorem which is of fundamental import- ance in the subject of the present volume. If a. be the instantaneous screw about which a quiescent rigid body either perfectly free or constrained in any manner whatever commences to twist in consequence of an impulsive wrench on some screw y, and if ß be another instantaneous screw, similarly related to an impulsive screw %, then whenever £ is reciprocal to a we shall find that y is reciprocal to ß. When this relation is fulfilled the screws a and ß are said to be conjugate screws of inertia. 82. The Determination of the Impulsive Screw, corresponding to a given instantaneous screw, is a definite problem when the body is perfectly free. If, however, the body be constrained, we shall show that any screw selected from a certain screw system will, in general, fulfil the required condition. Let By, ... B^n be 6 — to screws selected from the screw system which is reciprocal to that corresponding to the freedom of the nth order possessed by the rigid body. Let S be the screw about which the body is to twist. Let X be any one of the screws, an impulsive wrench about which would make the body twist about $; then any screw K belonging to the screw system of the (7 - n)th order, specified by the screws, X, By, ... B^n is an impulsive screw, corresponding to S as an instantaneous screw. For the wrench on Y may be resolved into 7 - n wrenches on X, By, ... B6_n-, of these, all but the first are instantly destroyed by the reaction of the constraints, so that the wrench on Y is practically equivalent to the wrench on X, which, by hypo- thesis, will make the body twist about 8. As an example:—if the body had freedom of the fifth order, then an impulsive wrench on any screw on a certain cylindroid will make the body commence to twist about a given screw. As another example:—if a body have freedom of the third order, then the “locus” of an impulsive wrench which would make the body twist about a given screw consists of all the screws in space which are reciprocal to a certain cylindroid. 83. System of Conjugate Screws of Inertia. We shall now show that from the screw system of the ?ith order P, which expresses the freedom of the rigid body, generally n screws can be selected