NEWTON S PEINCIPIA. 251 



This admits of a very short proof, for the equation of 

 motion being 



.ii +*** 



a s 

 we have by one integration 



and at the limits of integration s= a Q and s= + p we 

 have v = o 



which is what we had to prove. 



We have now to consider the nature of the curve a KB. 

 We have accurately true 



s = a sin (n t + b) 

 v = a n cos (n t + b) 



calling y the ordinate and supposing the resistance to be 

 equal to k v m 



Let x be the abscissa measured from the middle point O 

 of a B 5 and 



x + a i- a * = a s in ( n t + ft). 



JYr**, let the resistance vary as the velocity, then eli 

 minating t, 



If we neglect the variations of a with t, this is the equation 

 to an ellipse. The terms in y thus neglected are of the 

 order x 2 . Secondly, let the resistance vary as the square of 

 the velocity. Eliminating t 



