A Treatise on the Theory of Screws

169-172] FREEDOM OF THE THIRD ORDER. 171 six independent displacements. Its position is, therefore, to be specified by six co-ordinates. If, however, the body be so constrained that its six co- ordinates must always satisfy three equations of condition, there are then only three really independent co-ordinates, and any position possible for a body so circumstanced may be attained by twists about three fixed screws, provided that twists about these screws are permitted by the constraints. Let A be an initial position of a rigid body M. Let M be moved from J. to a closely adjacent position, and let x be the screw by twisting about which this movement has been effected; similarly let y and z be the two screws, twists about which would have brought the body from A to two other independent positions. We thus have three screws, x, y, z, which com- pletely specify the circumstances of the body so far as its capacity for movement is considered. Since M can be twisted about each and all of x, y, z, it must be capable of twisting about a doubly infinite number of other screws. For suppose that by twists of amplitude x', y', z', the final position V is attained. This position could have been reached by twisting about some screw v, so as to come from A to V by a single twist. As the ratios of x' to y', and z', are arbitrary, and as a change in either of these ratios changes v, the number of v screws is doubly infinite. All the screws of which v is a type form what we call a screw system of the third order. We may denote this screw system by the symbol >S'. 171. The Reciprocal Screw System. A wrench which acts on a screw y will not be able to disturb the equili- brium of M, provided y be reciprocal to x, y, z. If y be reciprocal to three independent screws of the system S, it will be reciprocal to every screw of S. Since y has thus only three conditions to satisfy in order that it may be reciprocal to S, and since five quantities determine a screw, it follows that y may be any one of a doubly infinite number of screws which we may term the reciprocal screw system S'. Remembering the property of reciprocal screws (§ 20) we have the following theorem (§ 73). A body only free to twist about all the screws of S cannot be disturbed by a wrench on any screw of S'; and, conversely, a body only free to twist about the screws of S' cannot be disturbed by a wrench on any screw of S. The reaction of the constraints by which the freedom is prescribed constitutes a wrench on a screw of S'. 172. Distribution of the Screws. To present a clear picture of all the movements which the body is