316 APPLIED SCIENCE 



independent of the magnitudes of the forces acting in the ele- 

 ments e and /, as was the case when we neglected weight and 

 inertia. The effort in e or in the resistance in / must therefore 

 be given, as well as the loads, before the frame can be drawn. 

 Let the effort in e or in link 1 be known, the frame and its re- 

 ciprocal may then be drawn as follows, so as to solve the pro- 

 blem of finding the resistance which a given effort in e will 

 overcome in /, neglecting the effect of friction, but taking into 

 account the weight and inertia of the parts. 



We know the position of four of the angles of the polygon, 

 viz., the centres of the pins at the four coiners of the machine. 

 Let the loads be referred to these four points in the manner 

 practised for the distributed loads on the actual rafters of a roof 

 or members of a bridge ; that is to say, let each load be replaced 

 by two components acting at these four points. These compo- 

 nents are lettered l a l' M l b I' ,etc. The stress in element /will be 

 the same as would be produced by the effort in e acting on the 

 quadrilateral frame abed, loaded at the joints in this manner. 

 This stress in / is found by drawing the Fig. 35a, beginning 

 with the polygon d, l d) la, l' a , a. The directions of all these 

 are known, and the magnitude of the stresses in all except a and 

 d. The polygon serves to determine these stresses. To find 

 the stress in b we require to draw the polygon a, l a , 6a, l' b , b, in 

 which there are only two unknown stresses those in 6a and b, 

 the directions of which are, however, given. We cannot draw 

 the polygon directly with the lines arranged in the manner 

 shown in full lines, because 6a and b are not contiguous. If, 

 however, the lines are drawn as dotted, we obtain a polygon 

 which determines the stress both in 6a and in b, after which the 

 lines may be rearranged in their natural order. By a similar 

 process we find 6/3 and c, and can complete and check the draw- 

 ing by adding the lines l' c , 1(3, and l b . It is almost unnecessary 

 to remark that la and 1/8 are equal, and that 6a and 6/3 are 

 also equal. The sides of the polygon in Fig. 35a represent the 

 loads on the elements of the machine in Fig. 35, taken in their 

 natural order. The lines abed in Fig. 35a represent the stresses on 

 the links abed, or 1, 2, 3 and 4 of the machine in Fig. 35. Let 

 o be the point where a, b, c, and d intersect (Fig. 35a), and let 

 lines be drawn from o to the angles of the polygon ; now, draw 



