14 BEST AND MOTION 



namely f^dt, f z dt. Then A B', A C' will represent the velocities at 

 the end of the interval dt. 



In the figure the lines BDF, B'ED', CDE, C'FD' are drawn 

 parallel to AB and A C. Thus AD represents the resultant velocity 

 at the beginning of the interval dt, and AD 1 that at the end of the 

 interval. The velocity AD 1 can be regarded as compounded of the 

 two velocities AD, DD', and, as in 10, DD' represents the incre- 

 ment in velocity in time dt. Thus, if F is the resultant acceleration, 

 the line DD' will represent a velocity Fdt. On the same scale DE, 

 DF represent velocities f^dt, f 2 dt, and DED'F is a parallelogram. 



If OF l} OF Z (fig. 8) represent the accelerations/!,/^ on any scale, 

 and if OG is the diagonal of the completed parallelogram, we 

 clearly have OF l : OF 2 = f^ :/ 2 = DE : DF, 

 so that the parallelograms OF^GF Z (in 

 fig. 8) and DED'F (in fig. 7) will be simi- 

 lar and similarly situated. Thus 

 OG : OF t = DD 1 : DE = Fdt :f^dt = F:f v 

 so that OG represents the acceleration F 

 on the same scale as that on which OF lf 

 OF 2 represent /p/2 ; and OG, being parallel 



to DD' , will also represent the direction of F, proving the theorem. 

 Clearly the acceleration at any instant need not be in the same 

 direction as the velocity. In fig. 7 the directions AD, AD' repre- 

 sent velocities at the beginning and end of the interval dt. When 

 in the limit we take dt = 0, these lines coincide, and the direction 

 of the velocity at the instant at which dt is taken is that of AD. 

 The direction of the acceleration at this instant is, however, DD'. 



As an illustration of this, let us consider the motion of a particle mov- 

 ing uniformly in a circle ; e.g. a point on the rim of a wheel, turning with 

 uniform velocity V about its center. 



Let A, B (fig. 9) be the positions of the point at two instants, let the 

 tangents at A , B meet in C, and let D complete the parallelogram A CBD. 



The velocity at the first instant is a velocity V along A C. Let us agree 

 to represent this by the line A C itself. At the second instant the velocity 

 is a velocity V along CB ; this may, on the same scale, be represented by 

 the line CB, or more conveniently by AD. Since AC, AD represent the 



