EQUATIONS OF MOTION 221 



These are the equations of motion of a particle in analytical 

 form. They simply express the second law of motion in mathe- 

 matical language. 



177. Let x t y, z be the coordinates of the particle at any instant, 

 and let u, v, w be the three components of its velocity. The com- 

 ponent u is the velocity, along the axis of x, of the projection of 

 the moving point on the axis of x, and the distance of this point 

 from the origin at any instant is simply <c. Thus, by the defini- 

 tion of velocity, we have 



dx 

 u = -, (67) 



and similarly, of course, v = -p> 



at 



dz 



w = 

 dt 



The rate at which the ^/-component of velocity increases is > 



at 



but it has also been supposed to be/ x , for this is the ^-component 

 of the acceleration. Thus we have 



du 



L= Tt' 



dv 



, 



at 



dw 



, . ., , 

 and similarly y 



at 



Using the values just found for u, v, w, these equations become 



ffx 

 fx *' 



df 



