244 NEWTON S PRINCIPIA. 



A particle constrained to move in a cycloid whose axis is 

 vertical, is acted on by gravity and resisted by the medium 

 in which it moves in the ratio of the velocity. To deter 

 mine the motion. Newt. xxvi. 



Let I be twice the radius of the generating circle. Let 

 s be the distance of the particle at any time t from the 

 lowest point of the cycloid, and v be the velocity, and m the 

 mass of the particle. Let x be the coefficient of resistance. 

 Then the moving force along the tangent will be 



- - . s - K v 



the particle being supposed ascending its arc. But this is 



d s g 



* 



also m -7- 2 . Hence 



m 



Tt 



is the equation to find s in terms of t. 

 To integrate this equation, put 



t*t 

 s = u . e 



where p is some constant at our disposal ; on substitution, 

 the equation becomes 



dp 



Let us then choose i*. so that 



2 p. + - = 0, 

 m 



and the above equation is reduced to the standard form 

 d*u 



V &quot; F \ 4m 1 



