NEWTON S PRINCIPIA. 191 



From these equations we learn by interpretation several 

 facts. 



1. Let the velocity of projection be in the direction in 

 which the accelerating force / acts. Hence f is here 

 positive. 



. Since e m can never change sign, the quantities 



fm , T7 fm 



v J ~ and V - 



X X 



have always the same sign. Hence the velocitv is always 

 greater or always less than - according as the velocity 

 of projection is greater or less than that quantity. 



(B. Since e ~ * continually diminishes as time goes on, 

 but never vanishes for any finite value of t, hence the ve 

 locity v continually approaches - , but never actually 



X 



equals it. 



y. If F the velocity at any one point should equal 



1 , the velocity is always equal to the same quantity. 



8. Since v continually approaches a finite quantity, the 

 expression for x shows that the space described continually 

 increases and finally becomes infinite in an infinite time. 



e. The velocity continually approaches the limit ^- *, 



hence x is the mass of that body whose limiting velocity 

 is/. 



2. Let the velocity of projection be in a direction op 

 posite to that in which the force/ acts. Here /is negative, 

 and our formula become 



fm 



v + J = 



X 



