510 PROFESSOR R. H. SMITH ON A NEW GRAPHIC 



is always varying throughout the periodic motion of the mechanism ; it is 

 always necessarily an awkward SGale to measure to, and it periodically becomes 

 in most cases an impossible scale by becoming infinitely large. Thirdly, the 

 various bars of a mechanism have all different instantaneous axes, and the scales 

 of the velocities would be entirely different for the different bars. As will be 

 shown presently, in the method explained in this paper, the velocity diagrams 

 of all the bars of even the most complicated mechanisms are all grouped 

 together so as all to radiate from one pole, and so as to be to the same scale for 

 all the bars and at all times throughout the periodic motion of the mechanism. 



A similar construction is applicable to accelerations of velocity. In fig. 2 

 let ABCD be one rigid bar. Let the acceleration of velocity of point A through 

 the field of the base-plate be known, and represented in direction and magni- 

 tude by p'a' drawn from any convenient pole p' to any convenient accelera- 

 tion scale. The acceleration of B can be obtained by adding to the vector p'a! 

 the vector acceleration of B in its relative motion round A. If &> be the 

 angular velocity of the bar in the base-plate field, and if J be the acceleration 

 of angular velocity, the radial or centripetal component of velocity acceleration 

 is w"AB and the tangential component is w'-AB. The whole acceleration of 

 relative velocity is therefore, ABV^ + w' 2 , and its direction is inclined to BA 



by the angle tan -1 u This angle is the same for every pair of points in the 



CO 



same rigid bar ; and, since the magnitude of the acceleration of one point round 

 any other is proportionate to the distance between them, therefore, if a'V be 



drawn inclined to BA at the angle tan -1 - 2 and of length AB^ + o/ 2 , and if 



the figure a'b'c'd' be made similar to that of the bar ABCD, then p'b\ p'c, p'd\ 

 will be the accelerations of the points BCD in the base-plate field. Further 

 ac, for instance, is the acceleration of C round A. In the graphic construction 

 it is simplest to plot «'j8' = AB -co\ and parallel to BA (not AB), and 

 /37/ = AB'6>' perpendicular to AB and in the direction given by the sign of at', 

 The radial component is usually obtainable from the already constructed 

 velocity diagram, where the velocity of B round A is called ab, and the radial 



acceleration is therefore ~~. Fig. 3 gives the two most ready graphic con- 



AB 



structions for calculating @£. In figs. (1) and (2) the velocity ab is plotted 



along BA from B to /3 towards A in (1) ; and away from A in (2). From B as 

 centre with B/3 as radius, a circular arc is struck intersecting in /3 2 any other 

 radius from A. From /3 is drawn /3/3' parallel to that other radius, and 



intersecting Bft in /3'. Then B/3' is the radial acceleration ^. In (3) (ab) is 



plotted from B as B/3 perpendicular to AB, and a circular arc with centre in 



