32 VELOCITY AND ACCELERATION. [CHAP. III. 



displacement in a unit of time. If the unit of time were replaced by a 

 smaller unit the displacement in it would be replaced by a shorter length, 

 and this length would measure the velocity in terms of the new unit of time. 

 However short an interval is taken for the unit of time the length described 

 in it measures the velocity in terms of it. When we wish to recall this fact, 

 and to bring it into connection with the definition of variable velocity we say 

 that the latter is measured by " the rate of displacement per unit of time," 

 but we must not attach to this phrase any other meaning than that which has 

 just been explained, i.e. the phrase means nothing but the limit of the fraction 

 number of units of length described in an interval 

 number of units of time in the interval 



when the interval is indefinitely diminished. 



30. Velocity in general. When the point is not moving 

 in a straight line it will have a displacement in any interval if t 

 parallel to each of the three axes of reference ; suppose these dis- 

 placements to be x' x, y' y, z' z. Then we assume that each 

 of the fractions 



x r x y' y z z 

 t'-t ' t' -t' t' -t' 



has a limit, and these limits are, as above, the rates of displace- 

 ment per unit time parallel to the axes. They are denned to be 

 the component velocities parallel to the axes. We can thus define 

 the velocity of a moving point in general to be a vector, localised 

 in a line through the position of the point, whose resolved part in 

 any direction is the rate of displacement of the point in that direc- 

 tion per unit time. 



As before, x, y,z are functions of t, and the component velocities 

 in the directions of the axes are 



dx dy dz 

 ~di' ~a~t y dt' 



At any instant the point is moving along the tangent to a curve, called its 

 path or trajectory. This tangent is the line drawn through the point in 

 the direction of the velocity, i. e. it is the limiting position of a chord drawn 

 from the point to an indefinitely near point of the curve. Let s be the arc of 

 the curve measured from some particular point of the curve up to the position 

 of the moving point at time t, and let s' be the corresponding arc for time t'. 

 Then the length of the chord joining the two positions is the magnitude of 

 the vector whose components parallel to the axes are #'-#, y'-y, s'-z, and 

 this chord becomes ultimately equal to the element of arc (ds) of the curve 

 a& the two positions approach to coincidence. Thus the magnitude of the 





