NEWTON S PRINCIPIA. 253 



1 



2 

 4 



8 



The difference between the arc of descent and the sub 

 sequent arc of ascent will be above differences divided by 

 the number of oscillations. In the greater arcs these are 

 very nearly as the squares of the arcs described, and in 

 lesser oscillations in a ratio somewhat greater. But theory 

 taught us that if the resistance vary as the m ih power of 

 the velocity, the above difference of arcs should vary as 

 the 7n th power of the arc. Hence the resistance when the 

 globe moves very swift is in the duplicate ratio of the 

 velocity nearly, and when it moves slowly, somewhat 

 greater than in that ratio. 



The points to be determined by experiment are how the 

 resistance depends on the velocity, how on the surface of 

 the body, and how on the density of the fluid. To deter 

 mine the first, we must assume some law of resistance : 

 let us suppose that the resistance varies partly as the 

 velocity, partly as the square of the velocity, and partly 

 in some intermediate ratio. This being the result sug 

 gested by the last-mentioned experiments ; let the differ 

 ence of the arcs be represented by 



A V + B V^ + C V , 



where V is the greatest velocity in any oscillation. Then, 

 according to the notation of the preceding theory, we 

 have 



hence substituting, the difference of the axes will be repre 

 sented by 



