94 THEORY OF NEWTONIAN FORCES. [PT. I. CH. I. 



and v z = vJ* + vJ + vJ 2 



We might have defined the vector velocity as the resultant of 

 the three vectors 



_ _dx - _dy _ _dz 

 dt ' dt ' dt 



50. Accelerations. If the velocity of a point is variable 

 with the time we define the acceleration of the point as the limit 

 of the ratio of the increment of velocity Ay to the increment of 

 time A, as both approach zero. We may consider either the 

 numerical change 



iV dv d 2 s 



or the geometrical change. If we draw a vector AB to represent 

 the velocity at the time t and the vector AC to represent the 

 velocity at the time t + A, and draw the arc of a circle BD, DC 



will represent the numerical change of 

 velocity, Av, not considering its direction, 

 while BG represents its geometrical, or 

 vector change, A#, for 



AS + BG = AC, 



FIG. 25. 



Accordingly Lim = Lim 



EG 



is the vector acceleration a. 



Since the projections of the geometrical difference of two 

 vectors are the differences of the projections, the components of a 

 in any direction will be proportional to the changes of the corre- 

 sponding components of the velocities, that is 



dv x d 2 x 



_ dv y 

 ~ 



dv z 

 dt 



