COMPOSITION OF FORCES 97 



original forces P, Q have been replaced by the new forces P', Q r . 

 The lines of action of these forces, however, will not in general be 

 parallel, so that they may be compounded into a single resultant 

 force acting through their point of intersection. 



73. Let us suppose that the forces originally to be compounded 

 were R lf R z ,--, and that these have been compounded into a 

 single resultant R. Let us take axes x, y in the plane in which 

 these forces act, and let the components of R 1 along these axes be 

 X^ Y lt those of R z being X v Y 2) and so on. Finally let the com- 

 ponents of R be X, Y. 



The system of forces which consists of the original forces R v 

 ^2 " "> together with the resultant R reversed, constitutes a system 

 in equilibrium. Resolving parallel to the axes, we obtain 



X\ + X, + X Z + ---- X=0, 



Y!+ r 2 + r 3 + ---- r=o. 



Thus the components of R are given by the equations 

 X=X l + Xt + X 9 + >-, 



Y= *!+ r a + y, + .... 



The magnitude of R can be found from the equation 



R 2 = x 2 + r 2 , 



while the angle 6 which the line of action of R makes with the 

 axis of x can be found from the equation 



To obtain the position of the line of action of R we use the fact 

 that the sum of the moments of 



R I} R z , R S) , and R 



taken about any point in the plane must vanish. This gives us 

 the moment of R about any point, and hence, since we know the 

 magnitude and direction of R, we can find the position of its line 

 of action. 



