A Treatise on the Theory of Screws

TWISTS AND WRENCHES. 7 1-3] which had an appropriate position in space, and an appropriate number of threads to the inch. In the Theory of Screws the word pitch is employed in a particular sense that must be carefully noted. We define the pitch of a screw to be the rectilinear distance through which the nut is translated parallel to the axis of the screw, while the nut is rotated through the angular unit of circular measure. The pitch is thus a linear magnitude. It follows from this definition that the rectilinear distance parallel to the axis of the screw through which the nut moves when rotated through a given angle is simply the product of the pitch (if the screw and the circular measure of the angle. 2. Definition of the word Screw. It is a fundamental principle of the theory developed in these pages that the dynamical significance of screws is precisely analogous to their kinematical significance. It is, therefore, essential that in the formal definition of the particular sense the word screw is to bear in this volume no prominence can be assigned to kinematical terms or conceptions unless it can be equally given to dynamical terms and conceptions. This condition is fulfilled by excluding both Kinematics and Dynamics and constituting the screw as the geometrical entity thus described. A screw is a straight line with which a definite linear magnitude termed the pitch is associated. We shall often denote a screw by a symbol, and then usually by a small Greek letter. With reference to these symbols, a caution may be necessary. If, for example, a screw be denoted by a, then a is not an ordinary algebraic quantity. It is a symbol which denotes all that is included in the conception of a screw, and requires five quantities for its specification ; of these four are required to determine the position of the straight line, and the pitch must be specified by a fifth. It will often be convenient to denote the pitch by a symbol, derived from the symbol employed to denote the screw to which the pitch belongs. The pitch of a screw is accordingly represented by appending to the letter p a suffix denoting the screw. For example, pa denotes the pitch of a and is an ordinary algebraical quantity. 3. Definition of the word Twist. We have next to define the use to be made of the word twist. A body is said to receive a twist about a screw when it is rotated uniformly about the screw, while it is translated uniformly parallel to the screw, through a distance equal to the product of the pitch and the circular measure of the angle of rotation.