INTRODUCTION. 



tion of the force are both manifestly invariable ; but this 

 is not the case either with the moment of the couple 

 or the aspect of its plane. 



The origin, however, can be always 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 component 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 it. 



The Canonical Form is Unique. 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. Suppose two lines possessed the pro- 

 perty, then if the force and couple belonging to one were 

 reversed, they must destroy the force and couple belong- 

 ing 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 act- 

 ing along these must be equal and opposite. The two 

 forces would therefore form a couple in a plane per- 

 pendicular to that of the couple which is found by com- 

 pounding 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 sim- 

 ple form, in which no arbitrary element is involved. 



