RESISTANCE PROPORTIONAL TO THE SQUARE OF THE VELOCITY. 23 



By means of the two values of x 1 and x 2 , we get 



/ 4 aA a 2 e 



(12) jc=-a 2 -ali ICOSTZ/ COS 2nt, 



\ o / o 



dx / 4\ . 



= an ( i ) s 



dt \ 3 / 



and 



dx / A(l\ , 



sm /H sm 2nt. 



It is seen that the first value of /, after zero, which annuls 



-.is / = - = T; from which it follows that the duration of the 

 dt n 



oscillation is the same as if the resistance did not exist. 



The deflection of the ascending half oscillation which succeeds 

 initial half oscillation, wil 

 expression for x ; its value is 



the initial half oscillation, will be obtained by making /=-in the 



n 



The following deflection will be the same in absolute value 



as to the degree of approximation adopted 



i6a 2 e 

 a 2 = a -_. 



The oscillations decrease therefore in arithmetical progression ; they 

 will vanish after a number p of oscillations given by the greatest 

 solution by a whole number, of the inequality 



We shall have the time /j of the descending half oscillation by 

 making ^ = 0, in the equation (12), which gives 



(4ae\ ac 



i -- I cos nL -- 

 3 / 3 



cos 



