204 NEWTON S PRINCIPIA. 



Y = J 



where ex. is the radius of the semicircle, whence, by a 

 simple substitution, we get 



3 x 



2 x a y 



so that the density of the medium at any point varies 

 as tangent of the angle a radius through the point makes 

 with the vertical. Newton also determines the law of 

 density when the particle describes an hyperbola with one 

 asymptote vertical, chiefly with the view of finding an 

 approximation to the curve which a particle will describe 

 in a uniformly resisting medium. This was a problem 

 which Newton was unable to solve, except in this imper 

 fect and indirect manner. We shall not therefore dwell 

 on this, but will proceed at once to indicate the manner 

 in which the question is now answered. 

 Taking the equation 



d 2 y q z* s 



7 ~ m 



dx* V 2 cos 2 a * 



multiply both sides by VI + p*dx, where p = -r, and 



Cl OC 



integrate 



2 ~* T 2 f 



= H- cos 2 . \p V 1 -f p 2 



. K 2 a L-* r 



2^ 



\ 



In gunnery p is usually small, let us reject the powers of 

 p higher than the second, we get 



2-s 2x V 2 



g m =l-i- cos 2 a (tan a p) 



mcj l) 



