NEWTON S PRINCTPIA. 207 



dv k dt 



m 



for the sake of brevity, put 



a; 2 

 4 



and integrating, we get, after correction, 



a; 2 mfa 

 4 k &amp;gt; 



v + - - c 



c k 







this will enable us to find t in terms of u, and, by solving 

 the equation, v in terms of t. 



If the particle be moving in the direction opposite to 

 that in which f acts, and if, also 



the above integral becomes imaginary ; the true expression 

 will manifestly be, if 



b b am 



Again, we have 



d v x xv 2 



d x m m OL 



* _(** ^ m f*\ ^ 



h 2 U &quot; ~rv 



for the sake of brevity put 



