DAMPING. 223 



Equation (17) being put in the form 



if we replace p by u +y, y being a very small quantity, and expand 

 by Taylor's theorem, taking only first terms, arid observing that 

 (j>(u) = o, we obtain for y the approximate value 



L uu i _ 



R^>'() R 2 + e + S R 

 Putting 



(18) 



we obtain for values of /o 2 and 



If we take as origin of the time the period at which the needle 

 passes through an elongation of amplitude a, the constants A lt A 2 , 

 and A 3 will be defined by the condition that when /=0, we have 



x = a, = and 1 = 0. 



at 



The coefficient A 1 is proportional to the cube of the ratio , 



Jx 



and the corresponding time may be neglected. / being the time 

 of passage through the position of equilibrium, the value of x may 

 then be again put in the form 



* / t 



x = Ae~ et sin y(t- / ) = Ae~r * sin TT - , 



