8 BEST AND MOTION 



the frame has a velocity v 2 in a direction AC relative to the 

 second frame. Then in time dt the moving point describes a dis- 

 tance v^dt, say the distance AD, along AB relative to the first 

 frame, while the frame itself describes a distance v 2 dt, say AE, 

 along AC relative to the second frame. Let AF be the diagonal 



of the parallelogram of which AD, AE 

 are two edges; then AF will be the 

 resultant motion of the point in time 

 dt relative to the second frame. Since 

 the moving point describes a distance 

 AF in time dt, the resultant velocity 



11 i AF 

 will be - 



dt 



Let us now agree that velocities are 



to be represented by straight lines, the direction of the line being 

 parallel to that of the velocity and its length being proportional , 

 to the amount of the velocity, the lengths being drawn according 

 to any scale we please ; for example, we might agree that every 

 inch of length is to represent a velocity of one foot per second, in 

 which case a velocity of three feet a second will be represented by a 

 line three inches long drawn parallel to the direction of motion. 



In fig. 3 let Ap, Aq represent the velocities v v v l drawn on any 

 scale we please. Since the scale is the same for both, we have 



Ap : Aq = v 2 : v r 

 Now AE = v z dt, AD = v^dt, so that 



and hence Ap : Aq = AE : AD. 



If we complete the parallelogram Aprq, the diagonal Ar will pass 

 through F, and we shall have 



If V is the resultant velocity, it has already been seen that 



r-^. 



dt 



