194 



NEWTON S PRINCIPIA. 



, sin = ^/ = 



dt 



m 

 x 



^ . 



V sin a + 



a m\ --t 

 + ^ ) e * 

 x / 



m 



(3). 



When any one of the five quantities x, y, v, 0, t, are 

 given, these four equations determine the other four. It 

 is therefore reduced to be a mere matter of arithmetic 

 calculation to determine the position of the particle at any 

 time. It may be laborious and tedious, but there is no 

 difficulty in it. 



We shall now trace the curve the particle describes. 

 Find t from the second equation and substitute in the 

 fourth, we have 



y = 

 9 



+ - 



- (4.) 



Let O be the origin, O B the direction of projection, 

 A O C a horizontal. Take O C 



= . V cos a, and draw C B ver- 

 x 



tical. Then from the above equa 

 tion it is manifest that B C is an 

 asymptote to the curve. Take 



\ Q 



B A = ( J a, a quantity, it will 



be observed, that is independent 

 both of V and a. Join O A, and 

 let ft = L A O C 



. . tan ]8 = tan a + . ^ . 



V COS a 



- (5.) 



