22 FORCES IN THE SAME PL AXE. [CITAP. H, 



new forces, and whose opposite apex lies in the other force,, is 

 constant; or when the product of the intensity of the forces 

 into their perpendicular distance remains the same. The di- 

 rection of rotation, of course, must also remain the same. 



We shall see further on the significance of this area, or of 

 this product so much is clear, that a couple (or infinitely small, 

 infinitely distant force) is completely determined in its plane 

 when the direction of rotation is given, and the area of the tri- 

 angle or value of the product to which it is proportional, is 

 known. The couplo itself can be replaced by any two parallel 

 equal and opposite forces whatever, if only the triangle having 

 one force as base, and the opposite apex in the other, has a given 

 constant area.* 



22. Force and Equilibrium Polygons for any Number 

 of Forces iu a Plane. 



In PI. 3, Fig. 12 (I) we have the forces P^ acting in various 

 directions and at different points of application. P 2 and P 3 

 form a couple; that is, are equal, parallel, and opposite in di- 

 rection. Required the position, intensity arid direction of action 

 of the resultant. 



First, form the force polygon, Fig. 12 (a), by laying off the 

 forces to scale one after the other in proper direction. Thus 

 we have 1, 1 2, 2 3, 3 4, 4 5 in Fig. 12 (a) parallel respec- 

 tively to PX P 2 P 3 , etc., in Fig. 12 (b). The line necessary to 

 close the polygon, 5, is the resultant in intensity and direc- 

 tion. In intensity because the length of 5 taken to the scale 

 of force, gives the intensity of the resultant ; in direction 

 because acting from 5 to it produces equilibrium, while act- 

 ing in the opposite direction, from to 5, it replaces the forces. 



We have, therefore, only to find the position of the resultant 

 in the plane of the given forces in Fig. 12 (b). Hence : 



Second, choose anywhere a "pole " as C, and draw the lines 

 or rays, or " strings " S S t S 2 S 3 S 4 , etc. S and S 5 are evi- 

 dently components of the resultant, since they form with it a 

 closed figure in the force polygon. 



Third, form the equilibrium polygon abcdeo', Fig. 12 (J), 

 as follows : 



Draw a line parallel to S intersecting P t (produced if noces- 

 sary) at any point as a. From this point draw a line parallel 



* El'mente der OrapMschen Statik. Bauschinger. Mttnchen. 1871. Po, 

 11, 12. 



