208 NEWTON S PRINCIPIA. 



and integrating we have 



v + ~2 + C 2 x 



c 



*+ 2 * 



where the quantity C is obviously equal to what the left 

 hand side of this equation becomes when V, the initial 

 velocity, is substituted for v. 



If the particle move in the direction opposite to that in 

 which / acts, and if 



this expression becomes imaginary. It is obvious, how 

 ever, that if we put 



2 = f _ ^ 



that the true integral will be 



a 2 



r\ 



-.- 



where C is obviously equal to what the left hand side 

 becomes when V is put for v. 



The quantity x in these formula is the mass of that 

 particle whose terminal velocity is 



/? 



A/4 



the quantity , therefore, represents a number, thus we 

 see that the preceding expressions are perfectly homo 



geneous. 



In exactly the same manner we may proceed to de 

 termine the motion of a particle in a medium resisting ac 

 cording to any other function of the velocity. 



