I37-] THEORY OF COUPLES. Si 



allel and equal to the original single force P, and has the distance 



itfi i$ pllf-vv i 



from it. 



Hence, a couple and a single force in the same plane are 

 together equivalent to a single force equal and parallel to, and of 

 the same sense with, the given force, but at a distance from it 

 which is found by dividing the moment of the couple by the 

 single force. 



136. Conversely, a single force P applied at a point A of a 

 rigid body can always be replaced by an equal and parallel force 

 P of the same sense, applied at any other point A' of the same body, 

 in connection with the couple formed by P at A and P at A'. 



137. The proposition of Art. 135 applies even when the force 

 lies in a plane parallel to that of the couple, since the couple can 

 be transferred to any parallel plane without changing its effect. 



If the single force intersects the plane of the couple, it can 

 be resolved into two' components, one lying in the plane of the 

 couple, while the other is at right angles to this plane. On 

 the former component the couple has, according to Art. 135, the 

 effect of transferring it to a parallel line. We thus obtain 

 two non-intersecting, or skew, forces at right angles to each other. 



Let P be the given force, and let it make the angle with the 

 plane of the given couple, whose force is F and whose arm 

 is/. Then/* sin a is the component at right angles to the 

 plane of the couple, while P cos a combines with the couple 

 whose moment is Fp to a force Pcosa in the plane of the 

 couple; this force Pcosa is parallel to the projection of P on 



the plane, and has the distance from this projection. 



Pcosa 



Hence, in the most general case, the combination of a single 

 force and a couple can be replaced by the combination of two 

 single forces crossing each other at right angles ; it can be 

 reduced to a single force only when the force is parallel to the 

 plane of the couple. 



PART II 6 



