4 TWISTS AND WRENCHES. 



tion between two positions of a point is most simply 

 presented by associating the purely metric element of 

 length with the purely geometrical conception of a di- 

 rected straight line. In like manner the relation of two 

 positions of a rigid body can be most simply presented 

 by associating a purely metric element with the purely 

 geometrical conception of a screw, which is merely a 

 straight line, with direction, situation, and pitch* 



3. Instantaneous Screws. Whatever be the move- 

 ment of a rigid body, it is at every instant twisting 

 about a screw. For the movement of the body when 

 passing from one position to another position indefi- 

 nitely adjacent, is indistinguishable from the twist about 

 an appropropriately chosen screw by which the same 

 displacement could be effected. The screw about which 

 the body is twisting at any instant is termed the instan- 

 taneous screw. 



4. Definition of the word Wrench. It has been proved 

 in the Introduction, that the canonical form of a sys- 

 tem of forces acting upon a rigid body consists of a 

 force and a couple whose plane is perpendicular to the 

 force. We now introduce the word wrench, to denote a 

 force and a couple in a plane perpendicular to the force. 

 The quotient obtained by dividing the moment of the 

 couple by the force is a linear magnitude. Everything, 

 therefore, which could be specified about a wrench is de- 

 termined (if the force be given in magnitude), when the 

 position of a straight line is assigned as the direction of 

 the force, and a linear magnitude is assigned as the quo- 



* Those acquainted with the language of the Quaternions, invented by the 

 late Sir W. R. Hamilton, will perceive that a twist bears the same relation to 

 a rigid body which a vector does to a point ; each just expresses what is 

 necessary to transfer the corresponding object from one given position to 

 another. 



