1-3] TWISTS AND WRENCHES. 7 



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 of 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, p a 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 tiuist. 



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. 



