6, 7] HODOGRAPH. RADIAL AND TRANSVERSE VELOCITY. H 



vector with respect to a point taken as origin equal to the vector 

 velocity of the moving point. Thus the radius vector of any point 

 on the hodograph is parallel to the tangent at the corresponding 

 point of the path. If X, Y, Z are the coordinates of a point on 

 the hodograph, we have for the relation between the two curves, 

 1Q x ^ dx v dy 7 dz 



z ."--ar Y=== dt' z= di' 



so that having established the correspondence of point to point, we 



obtain the time from 



ds 



20) <-/- 



y 



We shall call any vector which is related to another vector as the 

 vector X, Y, Z is to the vector x, y, z, the velocity of the vector, 

 and by a natural extension, shall call the locus of the end of the 

 second vector drawn from a fixed origin the hodograph of the first 

 vector. Thus we call Hamilton's hodograph the hodograph of the 

 position vector of the first point. 



7. Polar Coordinates. If a point moves in a plane it may be 

 convenient to specify its position by means of polar coordinates. 

 Let r be the distance of the point 

 from the origin 0, cp the angle that 

 the radius vector makes with a 

 fixed line through the origin. If 

 now the point moves from A to 

 B (Fig. 4) in the time A, describ- 

 ing the space As, so that r turns 

 through the angle Aqp, at the same O 

 time increasing by Ar, we may re- Fig. 4. 



solve the velocity into two com- 



ponents, one proportional to AC, where AC is perpendicular to OB, 

 the other proportional to CB. We have then the following vector 

 equation 



,. (AC ~ 

 ^ 



. 



or, passing to the limit, 



<rn\ dm . dr 



21 ) v = r + di- 



The two components of may be called the radial velocity, 

 v r = -- > and the transverse velocity, v (p = r -~r The rate of increase 



of the angle qp is called the angular velocity ~- The vector equation 

 21) gives rise to the scalar equation 



