34 VELOCITY AND ACCELERATION. [CHAP. III. 



the limit of the fraction 



number of units of velocity added in an interval of time 



number of units of time in the interval 



when the interval is indefinitely diminished. The number v is a 

 function of the number t, and its differential coefficient with 

 respect to t is the acceleration, i.e. the acceleration is measured 

 , dv 



^Tf 



When the point is not moving in a straight line it will in 

 general have a variable velocity parallel to each of the lines of 

 reference (coordinate axes). Let u, v, w be component velocities 

 parallel to these axes at time t, and u , v , w corresponding com- 



./ ,, ,1 f ,. u u v v w w 



ponents at time t , then the fractions -; - , -, , , , - are 



t t t t t t 



assumed to have limits when the interval t t is indefinitely 

 diminished, and these limits are the differential coefficients 



du dv dw _, i i i , i 



~di di ~di vector which has these components parallel 



to the axes is defined to be the acceleration of the point, or in 

 other words we define the acceleration of a moving point to be the 

 vector, localised in a line through the point, whose resolved part in 

 any direction is the rate of increase of the velocity in that direction 

 per unit of time. 



33. Measurement of Acceleration. The measure of any 

 particular acceleration is the number expressing the ratio of the 

 acceleration to the unit acceleration. 



The unit acceleration is that uniform acceleration with which 

 a moving point gains a unit of velocity in a unit of time. 



The number expressing an acceleration is the ratio of a number 

 expressing a velocity to a number expressing an interval of time. 

 It therefore varies inversely as the unit of length and directly as 

 the square of the unit of time. 



Acceleration is accordingly said to be a quantity of one 

 dimension in length and of minus two dimensions in time, or 

 its dimension symbol is LT~ 2 . 



34. Notation for velocities and accelerations. We 



have so frequently to deal with differential coefficients of 

 quantities with regard to the time that it is convenient to use 



