188 NEWTON S PRINCIPIA. 



integrating throughout the motion 



i v K i 



log ^ = t 



O V *vi 



m 



= V.. - - - - (4.) 



That is, when the times are in arithmetical progression, 

 the velocities are in geometrical progression. Also, we 

 have already proved that 



v-V = -- x - - - (5.) 



m 



or the velocity lost in passing over any space varies as 

 that space. 



As soon as the value of K is known, the above formulae 

 may be submitted to accurate calculation. As its value 

 depends on the form of the body, and the density of the 

 medium, it can only be found by experiment in any par 

 ticular case. 



We may, however, learn some curious facts from these 

 formulas. From the formula for v, in terms of t, we see 

 that though v continuously decreases as t increases, yet 

 it never vanishes. The particle will then never stop, 

 though constantly retarded. A little consideration will 

 show that this is just what we should expect. For the 

 resistance, varying as the velocity, takes away from the 



K d t 

 velocity in any small time d t, a certain fraction of 



the velocity that the particle has left. And as by taking 

 away continually the halves of any quantity no one can 

 remove the whole, so neither can this resistance ever 

 destroy the whole velocity. 



From the second formula we learn, that since v can 



