28, 29] DEFINITION OF VELOCITY. 31 



For a point moving in a straight line we may define the 

 average velocity in any interval of time to be the fraction 



number of units of length described in an interval 

 number of units of time in the interval 



When the point is not moving uniformly this fraction is a 

 variable number, which has a definite value when the measure of 

 the interval is given and the first instant of the interval is given. 

 Taking the first instant of the interval always the same, and 

 taking for the measure of the interval a series of diminishing 

 numbers, we obtain a series of fractions, which we assume* 

 approach a limiting value as the measure of the interval is 

 indefinitely diminished. This limiting value is defined to be the 

 velocity of the point at the first instant of the interval. We 

 might in the same way define the velocity of a point at the last 

 instant of an interval. 



We can now define the velocity of a point moving in a straight 

 line at any instant. It is the limit of the average velocity in an 

 indefinitely small interval of time beginning or ending at the 

 instant. 



The two limits are in general the same ; when they are differ 

 ent we call them the velocity just after the instant and the 

 velocity just before the instant respectively. 



Let t be the measure of the interval of time which has elapsed 

 since some particular instant, chosen as the origin of time, and 

 suppose that at the end of this interval the point has described a 

 length s measured from some particular point in the line of its 

 motion. We say that the point is at s at time t In the same 

 way suppose that it is at s at time t . Then in the interval t t 

 it describes a length s s, and its average velocity in the interval 



s s 

 is 77 - . The number s is a function of the number t, and the 



t ~~ t 



limit of the fraction just written is the number known as the 

 differential coefficient of s with respect to t. The velocity of the 



ds 



moving point is accordingly measured by -j- . 



at 



The number 5 - s is the measure of the displacement of the point during 

 the interval t -t. When the velocity is uniform it is measured by the 



* We do not require to consider any other case. 



