THE INERTIA OF THE CONNECTING ROD — MACGREGOR. 199 



the axes of crank positions and of velocities respectively. Then 

 MQ and JS^Q are the values of dc and dv for a small displace- 

 ment of the rod from the crank position A ; and when the 

 displacement is made indefinitely small, NQjMQ becomes ulti- 

 mately the dvjdc of the above formula, and MN becomes a straight 

 line. From M draw MR a normal to the curve at M. Then, 

 since MN, NQ and QM are perpendicular to MR, RA and AM 

 respectively, the triangles MNQ and MRA are similar. Hence 



AR^AM^ = v'^. 

 MQ dc 



From AL cut oft' a part AT equal to AR. Then T is a point on 

 a curve whose ordinates represent on some scale to be determined, 

 the values of vdv/dc for all crank positions. Other points may 

 be similarly determined, and a smooth curve drawn through them. 

 Its form is roughly indicated in Fig. 2 by the curve f. The dv 

 and dc at the crank position A, being both increments, and 

 vdv/dc being therefore positive, AT is drawn upwards. Between 

 A I and A2,dv isa, decrement ; vdv/dc is thus negative ; and hence 

 the ordinates of ^' are there drawn downwards. Obviously the 

 lines MQ and NQ do not require to be drawn in making the 

 construction. They are introduced above for purposes of proof. 

 Nor does MR require to be drawn. It is necessary only to 

 mark its end point R. If the drawing be made on co-ordinate 

 paper, the line MA will be a line on the paper. 



The mass m being a constant, S' — S is proportional to vdv/dc, 

 and its values in dififerent crank positions will therefore be repre- 

 sented by the same straight lines which represent the values of 

 vdv/dc. Hence the curve .^ gives not only the values of vdv/dc, 

 but also, if read to the proper scale, the values of S' — S for 

 all crank positions. If this scale be determined therefore the 

 problem is solved. 



There are four steps in the above method of obtaining values 

 of S' — S. First the curve VV is drawn from data of the problem. 

 In order to draw it, scales of velocity and of distance or length 

 must first be selected ; which means that we must select con- 

 venient units of velocity and length for the purposes of our 



