CHAP. H.] DIFFERENT POINTS OF APPLICATION. 23 



to S t to intersection with P 2 (also produced if necessary) at b. 

 From b parallel to S 2 to c, then parallel to Sg to d, and finally 

 parallel to S 4 , to intersection e with P 5 . Through this last point 

 draw a line parallel to the last ray S 5 . Now S and S 5 are com- 

 ponents of the resultant 5 [Fig. 12 ()] and are found in 

 proper relative position. Produce them, therefore, to intersec- 

 tion o'. Through this point the resultant must pass. Drawing 

 then through 6>', a line parallel to 5, we have the resultant in 

 proper position, and acting in the direction indicated in the fig- 

 ure, it produces equilibrium. 



Any other point than a, upon the direction of P l5 assumed as 

 a starting point, would have given a different point 0' ; so also 

 for any other assumed position of the pole C. But in every 

 case we shall obtain a point upon the line of direction of R^ 

 already found. The reader may easily convince himself of this 

 by making the construction for different poles, and points of 

 beginning. 



Now the polygon or broken line, a b c d e, we call the equi- 

 librium polygon that is, it is the position which a system of 

 strings or struts, S S x S 2 , etc., would assume under the action 

 of the given forces at the assumed points of application. 



Thus P! acting at , is held in equilibrium by the forces along 

 S and S t , P 3 acting at b, by S x and S 2 and so on. If we join 

 any two points in the line of direction of !? , and S 5 , as m n by 

 a line, we have then a jointed f/'ame, which acted upon at tlje 

 apices a. . .e by the forces P L . . .P 5 , and at m and n by S and 

 S 5 is in equilibrium. 



For S acting at m, we see from the force polygon may be 

 replaced by a force a parallel and opposed to the resultant R 

 and a force C a acting along the line L. In like manner S 5 may 

 be replaced by a C and 5 a parallel and opposed to the result- 

 ant. The two forces a C and C a being equal and opposed 

 balance each other through m n, while the sum of a and 5 a 

 is equal and opposed to the resultant 5. There is, therefore, 

 equilibrium, and m and n may be considered as the points of 

 support of the frame acted upon by the forces P lt . ,P 5 at the 

 apices a. . .e, a and 5 a being the upward reactions at the 

 points of support. 



As to the quality of the strains in the different pieces ; as 

 before the reaction at m, viz., a 0, is in equilibrium with the 

 strain in m n and m a. Following round, then, in the force 



