211.] 



CONDITIONS OF EQUILIBRIUM. 



125 



H is parallel to R, and has at the same time its least value, H Q ; 

 this value H is equal to the projection of any other vector H 

 on the direction of the resultant R. 



209. While, in general, a system of forces cannot be reduced 

 to a single resultant, it can always be reduced to two non-inter- 

 secting forces. This easily fol- 

 lows by considering the system 

 reduced to its resultant R and 

 resulting couple H for any 

 origin O (Fig. 69). Let F, 

 F be the forces, / the arm 

 of the couple //", and place this 

 couple so that one of the forces, 

 say F, intersects R at O. 

 Then, combining R and F 

 to their resultant F', the given 



Fig. 69. 



system of forces is evidently equivalent to the two non-inter- 

 secting forces F, F' (compare Art. 137). 



210. The two forces F, F' determine "a tetrahedron OABC\ 

 and it can be shown that the volume of this tetrahedron is con- 

 stant and equal to one sixth of the invariant of the system 

 (Art. 207). The proof readily appears from Fig. 69. The 

 volume of the tetrahedron OABC is evidently one half of the 

 volume of the quadrangular pyramid whose vertex is C and 

 whose base is the parallelogram OBAD. The area of this 

 parallelogram is Fp = H\ and the altitude of the pyramid is 

 = Rcos(j), being equal to the perpendicular let fall from the 

 extremity of R on the plane of the couple ; hence the volume of 

 the tetrahedron 



=RH cos 0= 



211. To effect the reduction of a given system of forces 

 analytically, it is usually best to refer the forces F and their 

 points of application P to a rectangular system of co-ordinates 



