Hg KINEMATICS. [217. 



-fcos0 =OQ.-OQQ=Q Q, if 2, (? are the projections of P, 

 P on the vertical AB. If we draw a horizontal line MN at 

 the height v/2g above P Q and if this line intersect the vertical 

 AB in R, we have for the velocity v the expression : 



(34) 

 If the initial velocity be =o, the equation would be 



(35) 



At the points M, N where the horizontal line MN inter- 

 sects the circle the velocity becomes o. The point can there- 

 fore never rise above these points. 



Now, according to the value of the initial velocity z/ , the line 

 MN may intersect the circle in two real points M, N, or touch 

 it at B, or not meet it at all. In the first case the point P 

 performs oscillations, passing from its initial position P Q through 

 A up to M, then falling back to A and rising to N, etc. In the 

 third case P makes complete revolutions. 



217. The second integration of the equation of motion cannot 

 be effected in finite terms, without introducing elliptic func- 

 tions. But for the case of most practical importance, viz. for 

 very small values of 6, it is easy to obtain an approximate solu- 

 tion. In this case 6 can- be substituted for sin#, and the 

 equation becomes : 



": ' + 7 =- '- < 36 > 



This is a well known differential equation (compare Art. 122, 

 Eq. (19), and Art. 176), whose general integral is 



sn 



6 = cos / 



