INTRODUCTION. 5 



perpendicular to the force. Thus a force, and a couple in a plane perpen 

 dicular to the force, constitute an adequate representation of any system of 

 forces applied to a rigid body. 



It is easily seen that all the forces acting upon a rigid body may, by 

 transference to an arbitrary origin, be compounded into a force acting at the 

 origin, and a couple. Wherever the origin be taken, the magnitude and 

 direction of the force are both manifestly invariable ; but this is not the case 

 either with the moment of the couple or the direction of its axis. 



The origin, however, can always be so selected that the plane of the 

 couple shall be perpendicular to the direction of the force. For at any origin 

 the couple can be resolved into two couples, one in a plane containing the 

 force, and the other in the plane perpendicular to the force. The first com 

 ponent can be compounded with the force, the effect being merely to transfer 

 the force to a parallel position ; thus the entire system is reduced to a force, 

 and a couple in a plane perpendicular to that force. 



It is very important to observe that there is only one straight line which 

 possesses the property that a force along this line, and a couple in a plane 

 perpendicular to the line, is equivalent to the given system of forces. Sup 

 pose two lines possessed the property, then if the force and couple belonging 

 to one were reversed, they must destroy the force and couple belonging to the 

 other. But the two straight lines must be parallel, since each must be parallel 

 to the resultant of all the forces supposed to act at a point, and the forces 

 acting along these must be equal and opposite. The two forces would there 

 fore form a couple in a plane perpendicular to that of the couple which is 

 found by compounding the two original couples. We should then have two 

 couples in perpendicular planes destroying each other, which is manifestly 

 impossible. 



We thus see that any system of forces applied to a rigid body can be 

 made to assume an extremely simple form, in which no arbitrary element is 

 involved. 



These two principles being established we are able to commence the 

 Theory of Screws. 



