36 Statics: 



If now in the equation 



pxFA + qxFB rxFC = gx FD, 

 found above, we put for p its value p + q r, just obtained, we 

 shall have 



p X FA + q x FB r X FC = (p + q r) X FD, 

 from which we deduce 



FD = p x FA + q x FB ~ r x FC 



p + q r 



or, bearing in mind, that the process by which we have arrived 

 at this result, does not depend upon the number of forces em- 

 ployed, we infer, as a general conclusion, that in order to deter- 

 mine at what distance from a given point the resultant of several par- 

 allel forces passes, from the sum of the moments of the forces which 

 tend to turn the system in one direction, we must subtract the sum 

 of the moments of the forces tending to turn it in the opposite direc- 

 tion, and divide the remainder by the sum of the forces which act in 

 one direction, minus the sum of those which act in a contrary direc- 

 tion.* 



67. If the point F, assumed arbitrarily, should happen to be 

 so taken as to fall in D, through which the resultant passes, the 

 distance FD being zero, its value 



p x FA + g X FB r x FC 



p + fif r 



since the force p tends to turn the system about the point D in a 

 direction opposite to that in which the force q tends to turn it, 

 becomes 



P x D ^ "*" 9 X DB rx DC 

 p + q r 



and is equal to zero ; we have consequently 



p x DA + q X DB r x DC = o, 



* We must take care not to confound the forces which act in op- 

 posite directions, with those which tend to turn the system in oppo- 

 site directions. Two forces which act in opposite directions often 

 tend to turn the system in the same direction. This depends upon 

 the point to which the rotation, or the moments, is referred. The 

 Fig. 21. two forces q, r, for example, act in opposite directions, but they 

 both tend to turn the line EC in the same direction, about a point 

 taken between B and C; and if we consider the rotation with refer- 

 ence to the point F, the force q tends to turn FC in a direction op- 

 posite to that in which the force r tends to turn it. 



