Art, 37. 



THE STRING POLYGON. 



43 



fall within the limits of the drawing. Two forces, 1 and 2, may 

 be introduced at any point A in the line of action of P lt such 

 that they will equilibrate Pi. This may be done by drawing any 

 convenient force triangle aOb and the lines through A parallel to 



1 and 2 (Oa and 60). The arrows within the triangle show the 

 proper direction for equilibrium. 



Where the line of 2 intersects P 2 (at B), introduce two forces 



2 and 3 such that they will equilibrate P 2 and one of them 

 be equal and oppos te to the force 2 used at P v These will 

 be the forces 2 and 3 from the force triangle bOc. In a like manner 

 P 3 and P 4 are equilibrated by the force 3 and 4 at C, and the 

 forces 4 and 5 at D from triangles cOd and dOe. There remains 

 yet unbalanced only forces 1 and 5 and therefore the equilibrant 

 of these must be the resultant of the whole system. It must 

 be equal to R of the force polygon and must pass through G, 

 the intersection of strings 1 and 5, in the space diagram. Since 

 R passes through G for complete equilibrium, E must pass through 

 G in an opposite direction. If E were parallel to R, as shown 

 in Fig. 28, the resultant would be a couple, because E=R in 

 the force polygon. 



The polygon ABCDGA is called a string polygon 1 ; its side* 

 are the strings, and these are drawn parallel to the components 

 1, 2, 3, etc. of the force polygon, which are called rays. The point 

 is called the pole. The last line drawn in the string polygon 

 is called the closing line, since it closes the polygon if all the 

 forces are in equilibrium. If the last and first strings do not 

 intersect on the line of one of the forces, there can not be equili- 

 brium, for they represent the lines of action of forces in the force 

 polygon, which form a triangle. Therefore, WHEN THE STRING 



POLYGON CLOSES, THE RESULTANT IS A SINGLE FORCE; WHEN THE 

 FORCE POLYGON CLOSES, THE RESULTANT IS A COUPLE; AND WHEN 

 BOTH POLYGONS CLOSE, THE RESULTANT IS ZERO THERE IS COM- 

 PLETE EQUILIBRIUM, THAT IS R = AND M = 0. 



The pole is so chosen as to give good intersections (not 

 acute) in the string polygon for locating the points B. C, etc. 



The above demonstration may be carried through by re- 

 solving each of the given forces into two components ; the arrows 

 on the rays will be "reversed. 



38. Properties of the String Polygon. The string polygon 

 represents only lines of action of forces whose magnitudes are 



1 Also called a funicular polygon and an equilibrium polygon. The 

 force polygon is also an equilibrium polygon. 



