202 NEWTON S PEINCIPIA. 



The resistance of the air being supposed to vary as the 

 square of the velocity and as the density conjointly) and the 

 forces to tend to directly to the plane of the horizon, to 

 determine what must be the law of density of the medium 

 that the particle may describe a given path, and to find the 

 velocity at any point. 



Let the axis of x be taken horizontal and that of y 

 vertical, let x, y be the co-ordinates of the particle at any 

 time t, and s the arc described. Let p be the density of 

 the medium at the point (#, y\ and v the velocity of the 

 particle, V the velocity, and a the angle of projection. 

 Then the resistance of the medium may be taken as 

 Resistance = K p v 2 . 



Let Y be the force acting on the particle parallel to the 

 axis of y. The equations of motion will be 



d*_x K 2 dx 



d~T 2 ~m pV Ts 



~ 



ds 



dt* m 



which may be put in the form 

 &quot; 



* = _ JL p v 1 - . (i.) 



7 &quot; m r c? * 



m f dt 



(t II fJ /* 



Multiply these equations by ~ and -7- and subtract, 



d x d 2 y _ d y d 2 x __ y d x ^^ 



dt d t 2 d t d t 2 &quot; d t 



By the theorem in the differential calculus for changing 

 the independent variable, we have, therefore, 



~d~x\ 2 d 2 y y 

 7Tt\ die 2 ~~ 



