APPLICATION OF GRAPHIC METHODS 283 



axes of elements d and /. The forces exerted by the elements d 

 and e on the element a must be balanced by the force due to 

 the third joint ab, therefore the direction of this balancing force 

 must pass through the intersection of 1 and 5, and must be 

 normal to the surface of the pin at ab. We therefore are now 

 able to draw the link 2. For similar reasons we must draw the 

 link 4 through the centre of the pin at c6, and through the inter- 

 section of the lines 5 and 6. The element & is in equilibrium 

 under the action of the four forces acting at the four joints ; in 

 other words, the resultant of 2 and 4 must be equal and opposite 

 to the resultant of 1 and 6, so that we may complete the dynamic 

 frame by drawing the link 3 as shown ; this last link, however, 

 would be equally well placed as shown in Fig. 56, where it joins 

 the intersection of 2 and 6 with that of 1 and 4. Both the 

 frames shown in Fig. ba or Fig. 56 are kinematically equivalent 

 to the actual machine shown in Fig. 5 in the following sense : A 

 given small contraction of the element e would, supposing all 

 the other elements to be rigid, produce a definite extension of 

 the element / in the actual machine. A small contraction in 

 link 1 equal to that in element e would, supposing all the other 

 links of the frame inextensible, produce an extension in link 6 

 equal to that produced in / in the machine. We may calculate 

 the relation between the stresses in e and / by the relative rates 

 of their contraction and extension, that is to say, by the principle 

 of virtual velocities, or we may calculate the relative stresses 

 between links 1 and 6 of the frame by the ordinary principles 

 of statics, for instance, by a ' reciprocal figure/ l The ratio 

 between the stresses in e and / and that between the stresses in 

 1 and 6 would be identical, whichever method were adopted. 

 It need hardly be said that the method by virtual velocities 

 would be much the simpler. It is not until we wish to take 

 friction into account that the utility of the dynamic frame 

 becomes apparent. In order to take friction into account we 

 have to change the form of the frame only in this respect, that 

 the links, instead of being normal to the surface of each joint. 

 must be inclined so as to make the angle of repose with the 

 normal to that joint, and must be so placed that the reaction due 



1 Vide J. Clerk Maxwell on Reciprocal Figures, Phil. Mag. t April 1864 ; 

 nd Fleeming Jenkin, Tram. Roy. Sec. Ed.,\ol. xxv. 1869 



