CHAP. I.] COMMON POINT OF APPLICATION. 5 



point to the beginning and end of any side of the force poly- 

 gon, and taking the direction of these lines opposed to the 

 direction of that side, we can decompose any force in the force 

 polygon into its components. Thus the force polygon gives ua 

 complete information as to the action of the forces. 



7. If tlie Forces act in the same straight lane, the force 

 polygon of course becomes a straight line also, and the result- 

 ant is the sum or difference (algebraic sum) of the forces. 



Thus, if we have P t , P 2 , P 3 , all acting at the point A, as 

 shown by the force diagram Fig. 4 (a), we form the force poly- 

 gon by laying off from 0, Fig. 4 (5), the intensity of P l5 from 

 the end of this line P t P 2 equal to A P 2 and from P 2 , P 2 P 3 

 equal to A P 3 . Then the line necessary to close the polygon is 

 evidently P 3 = P t + P 2 P 3 . A single force acting then at A 

 in the direction of and having the intensity represented by the 

 line P 3 would replace P x , P 2 , and P 3 . If acting from P 3 to 0, 

 it will produce equilibrium. 



If we again choose an arbitrary point as C [we shall hereaf- 

 ter call this point the "pole" of the force polygon], and draw 

 lines S S 3 from this pole to the beginning and end of the force 

 polygon, we can decompose the resultant into two forces in any 

 required direction. If the resultant is supposed to act down, 

 then the arrows show the direction in which these components 

 must act in order to replace the resultant. If then at A we 

 draw lines parallel and equal, we have these components in posi- 

 tion, direction, and applied at the common point of application. 



. Practical Applications. Simple and even self-evident 

 as all the preceding may seem, we have already acquired all 

 the principles requisite for a rapid, accurate, and very elegant 

 method of finding by diagram the strains in the various mem- 

 bers of all kinds of framed structures, such as roof trusses, 

 bridge girders, cranes, etc., no matter how complicated the 

 structure, or what special assumptions are made as to the load- 

 ing, provided only, that all the exterior forces are known. A 

 complicated or unsymmetrical arrangement of parts increases 

 greatly the labor of calculation, but has no effect upon the ease 

 or accuracy of the graphical method. The method moreover 

 checks its own accuracy, does not accumulate errors, and shows 

 in one view the relation of the strains to each other, and the 

 variations which would be caused by a change in the manner 

 of load distribution, or in the form of construction. 



