86.] INTRODUCTION. 47 



F at P', are in equilibrium; provided always that P and P ] 

 may be regarded as belonging to the same rigid body. 



85. It follows from Arts. 81 and 84 that any two forces F 1 

 F 2 whose lines intersect, say at O' (Fig. 1 5), are equivalent to, 



i.e. can be replaced by, a single force F called their resultant. 

 This resultant can be found by replacing the forces F v F% by 

 the equal forces F^, F 2 f at O f , and forming the parallelogram 

 having F^, F 2 ' as adjacent sides. The diagonal F' through O f 

 is the required resultant ; it can be replaced by any force F of 

 equal length and sense in the same line with this diagonal. 



The parallelogram construction need not be made at O' ; we 

 may select any origin O" (Fig. 15), draw through it two vectors 

 FJ' t F z " equal (in direction, length, and sense) to F v F 2 , find 

 the diagonal F 1 ' through O", and transfer it to a parallel line 

 drawn through O 1 . 



Finally, it is not necessary to draw the whole parallelogram ; 

 we have only to add the vectors F lt F 2 geometrically from any 

 origin O in (Fig. 15) and transfer their sum F" f to the parallel 

 through O'. 



86. Conversely, any force may be resolved into two com- 

 ponents along any two lines intersecting the line of the force 



