REDUCTION OF A SYSTEM OF FOl: 



perly placing this couple in its own j.lano? and makim: 



tfl forces ojual t<> /.'. one of i; may lie made t.> 



balance the force //. We shall then have remaining the 

 c.-ujile <7cos</> and a force 7.'. the direction of which is 

 j>mall<l t> the axis of the couple, and which is nx* 



a distance ^ from its original position. The system is 



thus reduced to a force 7? and a couple 



the axis of the latter being parallel to It, and therefore its 

 plane perpendicular to 7?. 



Since the resultant couple must !> independent of the dircc- 

 i of the axes of co-ordinates we conclude that 



B 



be constant whatever be the direction of the axes ; and 

 is constant it follows that 7.2 A' + J/ )"+ X2Z m>- 

 constant whatever be the direction of the axes. The expres- 

 sion also remains the same whatever ui-iyui be chosen, as ap- 

 pears from equation (2) of Art. 93. 



a system, of forces is reduced to a force and a 

 couple in a plane perpendicular to the force, the pu- 

 magnitude of the force are always the same. 



The magnitude of the force is always the same, for it is t lie 

 resultant of the given forces supposing each of tln-m moved 

 parallel to itself until they are all brought to act at the same 

 point. We shall now shew that there is a definite straight 

 fine along which the resultant force must act. 



Let ', y', z' be the co-ordinates of an origin such that the 

 of the resultant couple coincides with the direction of 

 the resultant force. Then, with the notation of Art. 93, 

 \\<- have 



L M' N 



for the direction cosines of the axis of the couple are propor- 

 1 to 7/, J/', and -V, and those of the direction of the 



