198 NEWTON S PKINCIPIA. 



the epoch at which the velocity is V. Also from (2) we 

 learn that when the spaces are in Arithmetical Progression 

 the velocities will be in geometrical Progression. 



A mere inspection of these equations shows that the 

 particle will continue to move for ever with a continually 

 decreasing velocity, and that it will pass over an infinite 

 space. 



We may represent, as Newton has done, the motion by 

 the several parts of an hyperbola. Construct the hyper 

 bola B P, whose asymptotes are the perpendicular straight 

 lines O X, O Y. Then P being any point in it and P N 

 an ordinate, we have 



ON. PN=c 2 



Take O A to contain ^ units of space, and A N to con 

 tain t units of space, then A N represents the time. 

 Since O N = t + ~ 



K C 



or the ordinate represents the velocity. 



/- 

 vdt 

 _ 



m 

 x c 2 



putting y for P N, hence the area A P represents the space 

 described. 



Secondly. We may proceed to consider the more 

 general case. We have 



dv K 



v -j- f tr 



dx m 



v d v 

 .-. =fdx 



