206 NEWTON S PRINCIPIA. 



d v K d t 



f a\ 2 a 2 &quot; 



(&quot; + 2) - 1 



integrating throughout the motion 



m 



So that when the times are in Arithmetical Progression, 

 quantities reciprocally proportional to the velocities, in 

 creased by a certain given quantity will be in Geome 

 trical Progression. This is Newton s eleventh proposition. 

 Again, since 



d v K K u 2 



dx &quot; m m a. 



d v it dx 



v + m a 



integrating throughout the motion 



_ 



.\ v + a = (V + a) e &quot; 



So that if the spaces described are taken in Arithmetical 

 Progression, the velocities augmented by a certain given 

 quantity will be in Geometrical Progression. This is 

 Newton s twelfth proposition. And by eliminating v be 

 tween the two equations, we can find x in terms of t. 

 But the result is complicated and of little value. 



Secondly. Let us proceed to the more general case : 

 we have 



dv / _ x . * J?. 2 

 d t ~~ ^ m m OL 



