186 NEWTON S 



formula, and we shall have an equation to determine the 

 motion. 



If x and y be the co-ordinates of the particle at any 

 time ^, it will follow by similar reasoning applied to the 

 second law of motion that the velocities parallel to the 

 axes are 



d x i d y 



~ and 



and the accelerations are 



parallel to the same axes. 



2. PROBLEM. To determine the motion of a particle moving 

 in a straight line in a medium, and thereby resisted in the 

 ratio of its velocity, and acted on by a uniform force. 



Let V be the velocity of the particle at any time which 

 we shall take as an epoch to measure our time from, and 

 the place of the body at this time, as our &quot; origin &quot; from 

 which to measure the distance of the particle at any 

 other time. Take as our direction of measurement the 

 direction in which the particle is moving at that epoch. 

 Let x and v be the distance and velocity of the body at 

 any time t. Let m be its mass, and let K v be the re 

 sistance when the body is moving with a velocity v ; that 

 is to say, let * v be the moving force which would be 

 required to keep the body moving, with this velocity in 

 the medium. Let / be the uniform accelerating force 

 measured in the usual way. 



