106 STATICS OF RIGID BODIES 



planes, the lines PS, P'S' being both perpendicular to AB. In the 

 same way let the couple AC be replaced by two forces QS, Q'S' 

 in these same two planes. 



The two couples have now been replaced by the four forces 

 PS, QS, P'S', Q'S'. 



Let us complete the parallelograms PSQR, BACD, P'S'Q'R'. 

 Obviously these parallelograms are all similar to one another, and 

 corresponding lines in the first and second parallelograms are at 

 right angles to one another. Thus a couple represented by AD 

 may be replaced by forces US, R'S f . But these two forces are 

 exactly equivalent to the four forces PS, QS, P'S', Q'S' to which, 

 as we have seen, the couples AB, AC may be reduced, and this 

 proves the theorem. 



FORCES IN SPACE 



83. When the forces acting on a body are not all in one plane, 

 their resultant will not in general be a single force. 



THEOREM. Any system of forces acting on a rigid body can be re- 

 placed ~by a force acting at an arbitrarily chosen point, and a couple. 

 Let G be the chosen point, and let R be any force of which the 

 line of action does not pass through G. At G let us introduce two 

 equal and opposite forces, each equal to R 

 and parallel to the line of action of R. By 

 combining one of these forces with the 

 original force R, we get a couple, so that 

 the original force R can be replaced by a 

 force parallel and equal to the original 

 force but acting at G, and a couple. 



Treating all the forces of the system in this way, we find that 

 the original system of forces may be replaced by 



(a) a number of forces acting at the chosen point G\ 



(b) a number of couples. 



The forces acting at G can be combined into a single force at G, 

 and the couples into a single couple, proving the result. 



