206 VI- SYSTEMS OF VECTORS. DISTRIBUT. OF MASS. INSTANT. MOTION. 



meet at E, if produced. Slide AC and BD to E, and then resolve 

 into components parallel to the original ones. We get EH and EJ 

 equal and opposite (being equal to AE and JBS), and EK equal to 

 AP and EL to BQ applied at E. Therefore the resultant of two 

 parallel forces is a parallel force equal to their algebraic sum, and 



applied on a line 



T EOj whose posi- 



tion is to be found 

 as follows. 



From the simi- 

 lar triangles, 



AO 

 FK 



BO 

 GL 



OE 

 ~KE 



OE 

 LE 



OE 

 AP' 



OE 

 BQ 



By division, since 

 FK=GL, 



Fig. 49. 



AO 

 BO 



BQ 

 AP 



Thus the position of the resultant of parallel forces is to be found 

 by the same construction as the resultant of two rotations about 

 parallel axes, Fig. 47. 



If the two forces are oppositely directed (Fig. 50), is on AB 

 produced, and if the forces are equal lies at infinity. Accord- 

 ingly there is 



no force that can 

 replace two equal, 

 parallel and op- 

 positely directed 

 forces not along 

 the same line, or 

 force- couple. The 

 distance between 

 the lines of direc- 

 tion is the arm, 

 and the product 

 of either force by 

 the arm is the 

 moment of the 

 couple. 



Fig. 50. 



We shall prove the following theorems. 



