ANALYSIS OF THE KINEMATICS OF MECHANISMS. 511 



BA is struck through A and /3. This arc cuts the diameter AB in /3' giving 

 B/3' the desired radial acceleration. 



If the bar be plotted to the scale m" = 1" , m being a fraction; and if the 

 velocity be plotted to the scale in!' = 1 inch/second ; then such constructions 



give the acceleration to the scale (— j inch = l inch/second. 2 



It is evident that the figure a'b'e'd' of the acceleration diagram is simply a 

 reproduction of the figure ABCD of the bar altered in scale and rotated through 



an angle -! 180° — tan -1 ^ [ in the direction of w where in tan -1 — 2 the sign of 



a' is to be taken positive or negative according as it is in the same or the 

 opposite direction to that of w. In this altered diagram p f is the point of the 

 bar if the bar extends so far, or of its field if it does not extend so far, which 

 suffers no acceleration or is moving uniformly in a straight line. This point does 

 not in general coincide with the instantaneous axis of rotation. If the velocity 

 diagram were rotated and altered in scale, and placed on top of the acceleration 

 diagram so that ab'c'd' coincided with abed, then p'p would represent in 

 direction and magnitude the acceleration of that line in the bar or in the field 

 of the bar which is at any time the instantaneous axis.* 



If G be the centre of inertia of the bar and the similar points g and g be 

 plotted in the velocity and acceleration diagrams, then the products of the 

 bar-mass by pg and p'g' are respectively the integral momentum and the 

 integral acceleration of momentum of the whole bar. 



In what follows the capital P will denote the base-plate through whose 

 field the velocities, &c, are reckoned. 



The pole of the velocity diagram will be called p. The pole of the accelera- 

 tion diagram will be called p. 



Points in the mechanism will be named by capital letters ABC, &c. 



The corresponding points in the velocity diagram will be named by the 

 same letters in small type, abc, &c. ; so that pc will denote the velocity of C 

 over the base-plate and be the velocity of C round B, and eb that of B 

 round C. 



The corresponding points in the acceleration diagram will be named by 

 accented small letters ; this being in accordance with the common mathematical 



convention, whereby x represents ~ 



In finding, for instance, the point b, it is sometimes necessary to find other 

 points which are not afterwards required in the completed diagram. When 

 such construction points require to be named, they will be called /3 U (3 2) &c, 



* The acceleration " image " of a bar moving without rotation reduces to a point. The velocity 

 " image " of a bar moving without rotation reduces to a point. 



