122 



STATICS. 



201. A further reduction is in general not possible. 

 general conditions of equilibrium are, therefore, 



[201. 



The 



202. Under special conditions it may of course happen that 

 R is perpendicular to the vector H. In this case R and H com- 

 bine to a single force R (Art. 135), and if the origin be taken on 

 the line of this force, the whole system reduces to a single 

 resultant. 



203. It is to be noticed that in the general reduction of forces 

 (Art. 200), the magnitude, direction, and sense of the resultant 

 force R are entirely independent of the position of the origin O, 

 the resultant being simply the geometric sum of all the given 

 forces. The resultant couple H, on the other hand, will in 

 general differ according to the origin selected. 



To investigate this dependence, let R t H (Fig. 66) be the ele- 

 ments of reduction for the origin O\ i.e. let R be the resultant, 



H the vector of the resulting couple of 

 a given system of forces when O is 

 selected as origin. To find the ele- 

 ments of reduction of the same system 

 of forces when some other point O' is 

 taken as origin, it is only necessary to 

 apply at O' two equal and opposite 

 forces R, R, each equal and parallel 

 to the original resultant R. The given 

 system of forces being equivalent to 

 R and H at O will also be equivalent 

 to the resultant R at O 1 , the couple 

 whose vector is H (which may be 

 drawn through O' without changing its effect), and the couple 

 formed by R at O and -R at O'. If / be the line of R through 

 O, V the line of R through O', and r the distance of these 

 parallels, the moment of the latter couple is Rr and its vector is 

 at right angles to the plane (/, O'). Combining the vectors H 



-R 



I 1 



Fig. 66. 



