196 THE INERTIA OF THE CONNECTING ROD — MACGREGOR. 



Hence the amount l)y which the required force is too great- 

 when calculated in this way, is given by the equation : 



S'-S ^'L(iJmk'). 

 dc 



This expression lends itself readily to graphical treatment.- 

 For this purpose we find w for various crank positions, by' 

 drawing diagrams similar to Fig. 1 for as many positions of the 

 crank as may be desired, measuring the lengths s in these posi- 

 tions and dividing the values of the velocity, T", of the pin for 

 these positions b}^ the corresponding values of s. We then plot a', 

 curve with distances traversed by the crank-pin from some- 

 initial position such as Aq, as abscissae, and the correspond- 

 ing values of w as ordinates, thus obtaining a curve which 

 gives us the values of w for all crank positions. Then selecting 

 points on this curve, whose ordinates have simple values, such 

 as can be raised to the square " in the head," or by reference to a 

 table of squares, we obtain a series of values of w^for the selectecT 

 crank positions ; and a second curve giving the variation of 

 J with the crank position may be plotted. Or as u' = u V/s,.. 

 we may obtain the u~ curve from the w curve by the con- 

 struction by which we obtain the curve vv from the curve VV^ 

 below^, V and s taking the place of the k and s used in that 

 construction. 



A similar curve for /r must next be obtained. As l^ = K'-\-cl-, its 

 values for different positions of the crank may be obtained by 

 finding the values of d from the diagrams similar to Fig. 1, 

 already drawn, and adding their squares to the scjuare of the 

 constant h. For this purpose draw a right-angled triangle- 

 abc (a diagram is not necessary), whose sides ab and he, contain- 

 ing the right angle, represent on any convenient scale the quan- 

 tities h and d. Then the hypothenuse ac wnll represent to the 

 same scale //j2~ip~p"or /.-. From a point e in ac at a distance 

 from a ore, say a, of one scale division, draw a line ef, in any 

 direction, e(iual to ac; join af, and through c draw a line parallel 

 to ^'/' and meeting a/' produced in g. The line c^ will represent- 

 /r + d? or A;"^ to the same scale as ah and he represent It and d. 



The I? curve and the k? curve must now be combined so as to 



