THE MOTION OF A SPHERE 191 



Putting w = ^ and integrating again with condition y = o when t = we get 



y = Y\t-^-{l-c-^t)\ (6) 



2. Horizontal motion icitli initial velocity H. 



If XI is horizontal velocity at time t, the equation of motion is now simply 



du 

 or 



Therefore 

 and hence 



(7) 



But X = X for u = 0, therefore H = cX. 



From the last expression and (3) we obtain 



H=f (8) 



Proceeding with the integration, from (7) we have 



dx 



-dr'' 



= B.-cx 



= c (X-x). 



Integration with initial condition x — o when t = o leads to 



a- = X (l-f-<') (9) 



3. The equation of the path of a sphere projected horizontally under gravity 

 is obtained at once by the elimination of t from the two equations (6) and (9) ; 



and replacing c by its value ^ we have finally 



