ANALYSIS OF THE KINEMATICS OF MECHANISMS. 509 



many mechanisms, he has so far failed to discover any practical use to which 

 these axoids can be applied. They are very tedious of construction, and when 

 constructed furnish no direct means of measurement of any useful quantities. 



The method now proposed furnishes velocity and acceleration diagrams, 

 somewhat similar in appearance to stress diagrams, which show the true 

 directions and magnitudes to scale of the velocities and velocity accelerations 

 of all points in the mechanism; there being one pole only for each diagram 

 from which all vectors radiate, so that the velocities or accelerations of all parts 

 and at all times of the complete cyclic period can be compared with maximum 

 facility. 



Fig. 1. — Let ABCD be a rigid bar. Suppose the velocity of A over the 

 base-plate P to be known. Choose any pole p, and drawjpct parallel to the 

 velocity of A, and of a length to represent its magnitude to any scale considered 

 convenient for the velocity diagram. If now the angular velocity <o of the bar 

 be also known, ab may be plotted perpendicular to AB, and equal in length to 

 (a- AB to the above velocity scale. Then, pb is the velocity of B over the base- 

 plate. If, instead of <o being known, we know the velocity of B as well as that 

 of A, then £>5 may be plotted directly, and joining ab the angular velocity may, 

 if desired, be calculated by dividing ab by AB. Since the (relative) velocity of 

 C round A is perpendicular to AC, and its relative velocity round B is perpen- 

 dicular to BC ; if ac and be are drawn perpendicular to AC and BC, their 

 intersection c gives pc the velocity of C through the base-plate field. Similarly, 

 pel is found to measure the velocity of D. The diagram gives not only the 

 velocities over the base-plate P, but also all the velocities of pairs of points 

 relative to each other. For instance, bd is that of D round B, and db is that of 

 B round D, these relative velocities being through the field of the base- 

 plate P. 



It is clear that the figure abed forms a diagram of the bar ABCD to a 

 diminished scale and turned through a right angle in the direction of a>* 

 Further, on this new diagram of the bar, altered in scale and rotated through 

 90°, the pole p represents the position of the instantaneous axis of rotation, 

 Theoretically, the original diagram ABCD, with the position P of the instan- 

 taneous axis added, would serve equally well as a diagram of velocities, the 

 scale being chosen suitably, so that PA would represent the velocity of A. 

 But for practical graphic construction it cannot be so used, for several reasons. 

 Firstly, the usual variation of the position of the instantaneous axis is extremely 

 inconvenient, and in almost all mechanisms this axis periodically recedes to an 

 infinite distance. Secondly, the scale to which it could represent the velocities 



* In an abstract of this paper written for the engineering journals, the late Professor Fleeming 

 Jenkin very expressively called abed the " image " of ABCD. In the acceleration diagram another 

 "image" a'b'c'd' appears. 



