NEWTON S PRINCIPIA. 229 



If the particles of a fluid become packed closer together, 

 so that the fluid becomes denser in the ratio of p to I, it 

 is manifest that the number of particles striking the sur 

 face will be increased in this ratio, and therefore the 

 resistance will be increased in the ratio of p to 1. If the 

 masses of the particles be increased in the ratio p to 1, the 

 density of the fluid will be increased in this ratio, the force 

 of each impact will be increased in this ratio, and the 

 resistance will be increased in the ratio of p to 1. Hence 

 the resistance varies as the density of the fluid. 



To show that the resistance on a sphere is half that on 

 one of its great circles. Let r be the radius of the sphere, 

 the angle any radius makes with the direction of the 

 motion. Then, by the general rule, the resistance on the 

 annulus 2 TT r sin. . r d & will be 



2 TT r 2 sin. d cos. 0, 3 . x u 2 . 

 Hence the whole resistance will be 



cos. 0| 3 sin. d 



*-, 

 4 



the limits of integration being from = to = -. 



2 



But the resistance on a circle of radius r is 



o o 



x v* . TT r , 

 which is just double the former result. 



These results were afterwards modified by Newton. A 

 course of reasoning, which we shall consider in another 

 chapter, led him to the conclusion that the resistance on a 

 sphere is equal to ! c p v 1 , where c is the area of one of its 

 great circles. 



4. Upon a circular base rad. r, construct afrustrum of a 

 cone of given height h, such that the resistance on it may be 



Q 3 



