200 THE INERTIA OF THE CONNECTING ROD — MACOREGOR. 



geometrical constructions. Let us suppose we select 9 inches 

 (equal to | ft.) to be represented by one division of the distance 

 scale, and 10 feet per second to be represented l)y one division of 

 the velocity scale. 



Secondly, the curve, vv, is obtained from the curve, VV ; and 

 as seen above the scales of the two curves are the same. 



Thirdly, the curve, Jf, is obtained from vv by applying a geo- 

 metrical construction, which is the equivalent of the algebraic 

 operation indicated by the expression vdv/dc. Hence the scale of 

 the ordinates of the curve /f' must be related to the velocity and 

 distance scales in the same way as the unit of vdv/dc is related to 

 the units of v and c. Now vdv/dc has the dimensions of an accel- 

 eration ; and the magnitude of a unit of acceleration is always 

 equal to the quotient of the square of the magnitude of the unit 

 of velocity by the magnitude of the unit of length', provided these 

 units are units of some one derived system, and their magnitudes 

 are expressed in terms of the same units of length and time. 



Hence our unit of acceleration must be 10"'-^--= — • feet-per- 



second per second. The scale of f, therefore, considered as 



giving values of vdu/dc for the various crank positions is — 



ft. -sec. units to a division. 



Finally, in employing f as a curve of force, we apply the 



equation : 



o' o dv 



o — >b = mv — , 



dc 

 without any further geometrical construction. It is olivious from 



400 



this equation that if vdv/dc have the value , »S" — >S' will have 



o 



the value . Hence the scale of the ordinates of jf, con- 

 sidered as a force curve, will be , the value of m varying 



with the unit of mass in terms of which the mass of the rod is 

 expressed. Also the above equation holds only provided all 

 quantities in it be expressed in terms of derived units. Hence 



