ISOCHRONOUS OSCILLATIONS. 25 



681. ISOCHRONOUS OSCILLATIONS. Let us first consider the case 

 of the imaginary roots. Putting y 2 = n 2 e 2 , equation (14) becomes 



x = e~ et [A l cos y/ + A 2 sin yi\. 

 We may write it in the equivalent form 



x = Ae~ el sin y (/ - / ). 



The constant / represents the period at which the movable body 

 passes through its position of equilibrium. If we count the time 

 starting from this passage, we have / = 0, and the equation reduces 

 to 



(16) x = Ae~ et sin yt ; 



from this follows 



= Ae~ ft (y cos yfe sin y/), 

 at 



which gives for the initial angular velocity 



The deflections correspond to the times for which we have = 0, 

 that is to say 



(17) tany/=2. 



The first takes place at the time / p given by the smallest root of 

 this equation; the next one at the time / 2 , such that 



that is to say, that it follows the first after an interval / 2 - / t , having 

 the value 



(.8) T = . 



The same is the case with all the others. The oscillations are then 

 still isochronous, but the time of an oscillation r is greater than the 



