10 THE THEORY OF SCREWS. [5- 



Although we have described the twist as a compound movement, yet in 

 the present method of studying mechanics it is essential to consider the 

 twist as one homogeneous quantity. Nor is there anything unnatural in 

 such a supposition. Everyone will admit that the relation 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 

 directed straight line. In like manner the relation between 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. 



It thus appears that a twist bears the same relation to a rigid body which 

 the ordinary vector bears to a point. Each just expresses what is necessary 

 to express the transference of the corresponding object from one given position 

 to another*. 



6. Instantaneous Screws. 



Whatever be the movement 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 indefinitely adjacent, is indistinguishable from 

 the twist about an appropriately chosen screw by which the same displacement 

 could be effected. The screw about which the body is twisting at any 

 instant is termed the instantaneous screw. 



7. Definition of the word Wrench. 



It has been explained in the Introduction that a system of forces 

 acting upon a rigid body may be generally expressed by a certain force 

 and a couple whose plane is perpendicular to the force. We now employ 

 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 determined (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 quotient just referred to. 

 Remembering the definition of a screw ( 2), we may use the phrase, 

 ivrench on a screw, meaning thereby, a force directed along the screw and 

 a couple in a plane perpendicular to the screw, the moment of the couple 

 being equal to the product of the force and the pitch of the screw. Hence 

 we may state that 



The canonical form to which a system of forces acting on a rigid body 

 can be reduced is a wrench on a screw. 



* Compare M. Rene de Saussure, American Journal of Mathematics, Vol. xvm. No. 4, p. 337. 



