APPLICATION OF GRAPHIC METHODS 317 



the lines 2a and 2/3 in Fig. 35, parallel to the lines of the same 

 name in Fig. 35a, and so placed that they abut against the ends 

 of link a, and intersect in the line L a ; draw 3a and 3/9, 4a and 

 4/J, ba and 5/3 by a similar rule. The octagonal frame thus ob- 

 tained, and shown by a broken dotted line, is a frame which will 

 be in equilibrium under the four given loads and the two stresses 

 in e and /. It is easy to see that this will be the case. The 

 load L a and the two stresses in 2a 2/2 of the frame form a 

 polygon in the reciprocal iigure. The same relation obtains 

 between the other loads and the links supporting them. More- 

 over, the links 'la and 5/3 in the reciprocal figure give a closed 

 polygon with the line la, representing the stress in e. The lines 

 abutting against the ends of/ also give closed polygons with the 

 stress 6, as shown in the reciprocal figure. The links 2, 2/3, 

 etc., of the octagonal frame do therefore represent the directions 

 of bearing pressures at the joints on the hypothesis that there is 

 no friction. The frame found by the method described will be 

 called the loaded dynamic frame without friction. 



The elements e and / have been represented as without mass ; 

 if their weight and inertia are to be taken into account, their 

 load is to be referred to the joints in a manner similar to th?t 

 indicated for the other elements. Links 1 and 6 would then be 

 broken lines in the frame, with loads at their angles. It may 

 be well to remind the reader that by hypothesis the machine is 

 in equilibrium as a whole, and therefore in the reciprocal figure 

 lines representing the loads necessarily form a closed polygon. 



28. Loaded Dynamic Frame ic/tJt Friction. The method by 

 which we were enabled to draw the octagonal polygon described 

 in the last paragraph depended on our knowledge of the four 

 points which determined the position of four angles of the 

 polygon, or one point on each of the eight lines. When we try 

 to ascertain the actual lines of bearing pressure, taking into 

 account the friction of the machine in motion, we find that the 

 conditions determining their direction are more complex, since 

 now we do not know any fixed point in any line. The conditions 

 are, however, only changed to this extent, that the lines of 

 bearing pressure must make the stated angle with the joints, 

 instead of being normal to the surfaces of those joints. By 

 trial, an octagon is easily found fulfilling this condition as well 



