508 PROFESSOR R. H. SMITH ON A NEW GRAPHIC 



more simply the " paths " of these parts. These paths are drawn in on the 

 " mechanism diagram." 



From these "paths" the displacements from any assumed initial configuration 

 can be directly measured. It may, however, be often advantageous to have a 

 separate " displacement diagram," consisting of a series of curves, showing the 

 successive simultaneous displacements of all important points of the mechanism 

 as vector-radii from one and the same pole. These curves in the " displace- 

 ment diagram " are, of course, exact copies of the " paths " in the " mechanism 

 diagram." 



Let ABC and BDE be two bars linked together at the joint B. Let P' be 

 the pole of the displacement diagram, and let the curves A' A', B'B', D'D' be 

 the displacement curves of the three points ABD. All these curves, of course, 

 pass through the pole P'. P'Ai, P'Bi, and P'Di, being simultaneous displace- 

 ments of AB and D, draw on A1B1 and B1D1 as bases, the triangles A1B1C1 and 

 BjDiEi similar to the triangles ABC and BDE in the mechanism. It can 

 easily be shown that P'Ci and P'Ei are the simultaneous displacements of C 

 and E, By joining all the points C and E' found by such constructions, the 

 displacement curves of C and E can be drawn in. Numerous simultaneous 

 points on the various displacement curves should be marked, and numbered 1, 

 2, 3, 4, &c. 



The advantage of such a displacement diagram over the set of " paths " 

 dispersed over the mechanism diagram, consists in the greater facility of com- 

 parison between the displacements of the various parts of the mechanism that 

 it affords. Thus, comparing simultaneous points A' and C belonging to the 

 same bar, the vector A'C is the displacement of C past (or relatively to) A in 

 the base-plate field. The same holds for points in different bars ; thus, D'C is 

 the displacement of C past D in the base-plate field. As a useful fact of 

 assistance in drawing these diagrams, it may be noted that any line, such as 

 A'C belonging to one bar, is perpendicular to the line bisecting the angle 

 between the simultaneous and "initial" positions of the line AC in the 

 mechanism. This does not, however, apply to a line, such as E'C, joining 

 points belonging to different bars. 



The method of obtaining the velocities by taking the small differences of the 

 displacements, which method is the basis of kinematic analysis developed by 

 means of the differential calculus, has often been adopted as a graphic process 

 for the solution of specially complicated problems. After constructing the 

 velocity hodographs, the same method may be followed to find the velocity 

 accelerations. As a graphic process, however, this method is capable of no 

 accuracy ; it is, in fact, practically useless. 



Professor Reuleaux's method of centroids, more properly called axoids, has 

 now become famous ; but, although the writer has constructed the axoids of 



