NEWTON S PRINCIPIA. 247 



Newton has given a geometrical construction for this 

 velocity. But it is very long, and in the present age, 

 when analysis is the great mathematical weapon, such a 

 complicated construction is of no value except as a matter 

 of curiosity. 



If we wish to have the means of deducing the law of 

 resistance from experiments on the pendulum, we must in 

 vestigate the changes produced in the time and arc of vi 

 bration by a resistance that varies according to any law, 

 indeed by any small disturbing cause whatever. The two 

 next propositions of Newton have this for their object. 

 They are entirely geometrical and the investigations too 

 complicated to be inserted here. Of one of them Newton 

 says : &quot; by reason of the difficulty by which the resistance 

 and velocity are found by this proposition we have thought 

 fit to subjoin the following.&quot; We shall first discuss the 

 modern analytical method* of determining the effect of 

 any small disturbing cause, and then give an analytical 

 proof of Newton s general proposition. 



Let the quantities s, t, n, &c. have the same meaning as 

 before, and let f be the small disturbing cause which acts 

 along the tangent on the particle. 



The equation of motion will then be 



dt* 



If/=o, the motion will be given by 



s = a sin. (n t + b) 



- = a n cos. (n t + 

 a t 



* Camb. Phil. Trans. 1826. 

 R 4 



