226 NEWTON S PRINCIPIA. 



This reasoning requires that the velocity should be so 

 great that the forces of the particles will not have time 

 to act. 



2. The preceding investigation has led us, on certain as 

 sumptions, to the law of resistance to similar bodies, but it 

 now remains to discover what change in the resistance 

 would be caused by a change of form in the body. A new 

 assumption becomes necessary. Let us suppose the par 

 ticles to be so rare that their distances are infinitely greater 

 than their diameters, so that each particle may be able to 

 give its blow to the body and then to make its escape 

 without affecting the particles which have not yet given 

 their stroke. It is manifest that to find the resistance 

 according to this principle we have to divide the surface 

 into elements, find the resistance on those elementary 

 planes separately, and by integration add the results. It 

 is necessary to find the resistance on a small plane in 

 clined at an angle to the direction of motion. Let the 

 area of the plane be A, then the number of particles that 

 will strike it will be proportional to A cos. and to the 

 velocity v conjointly, and each particle will strike the 

 plane with a normal velocity v cos. 0. The mass of each 

 particle is supposed the same. Hence the whole normal 

 resistance will be proportional to A cos. 0| 2 v 2 , and resolving 

 this along the direction of motion the resistance will be pro 

 portional to A~cosT0| 3 v 1 . Hence if K A v 2 be the resist 

 ance on the plane when perpendicular to the direction of 

 motion, the resistance when inclined at an angle will be 



x A v 2 cos. 0j 



It will be observed that this reasoning is true whether the 

 particles be elastic or not. Any change of elasticity 

 affects the resistance by changing x. 



Let a cylinder be made to advance in the direction of its 



