THE GEOMETRY OF MOTION 221 



the rays of the map of the path (Fig. 9, p. 21 o) give the 

 position of P relative to O, the rays of the hodograph 

 give the velocities of P relative to O. So soon as we are 

 in possession of the time-chart and the map of the path 

 we can construct this diagram of the velocities. When 

 constructed it forms an accurate picture of how the motion 

 is changing in both magnitude and direction. 



Let us now examine this hodograph a little more 

 closely. It consists of a point or pole I and rays IV 

 drawn from this pole to a curve V^ V„ V^ . . . V^^,. Now 

 this is exactly what the map in Fig. 9 consists of In 

 that figure we have a pole O and rays OP drawn from 

 this pole to a curve P^ P^ P3 . . . ^ ^^. In the course of 

 the motion P passes along the whole length of this curve, 

 and in just the same manner we may look upon V as 

 moving along the whole length of the hodograph-curve. 

 The ray IV would in each position be the displacement 

 of V relative to I. The question now arises : Has the 

 motion of V round its curve any meaning for the motion 

 of P in the path ? Suppose we were now to treat the 

 hodograph as the map of a new motion, and to construct 

 first the time-chart and then the hodograph of this motion, 

 what would the rays of this second hodograph represent ? 

 Now a sort of logical rule-of-three sum will give us the 

 answer to this question. As the rays of the first hodograph 

 are to the map of the path, so are the rays of the second 

 hodograph to the map of V's motion. But we have seen 

 that the rays of the first hodograph measure the velocities 

 of P in its path, and that these velocities are a fitting 

 measure of how the ray OP, or the position of P relative 

 to O, is changing. Hence it follows that the rays of the 

 second hodograph would measure the velocities of V in 

 the first hodograph, and that these velocities are a fitting 

 measure of how the ray IV or the velocity of P relative 

 to O is changing. Thus the velocity of V along the hodo- 

 graph is the measure of how the velocity of P relative to 

 O is changing. Tliis velocity of V, or change in the 

 velocity of P, is termed acceleration, and we see that a 

 diagram of accelerations may be obtained by drawing the 



