42 MEASUREMENT OF OSCILLATIONS. 



This method of correction obviously implies that the position of 

 equilibrium is not appreciably displaced during several vibrations. 



695. GAUSS'S METHOD. The following method, used by 

 Gauss,* may be applied to all kinds of vibrations, whatever be 

 the damping ; but it is especially convenient when the vibrations 

 are very slow. 



Suppose that we observe the image of a scale, the divisions of 

 which are numbered from one end. We shall call X Q , x v x 2 ---- the 

 divisions which correspond to the successive elongations (Fig. 133), 

 the even indices representing the oscillations to the left, and the odd 

 ones to the right. 



We must first determine the position of equilibrium by the 

 oscillations themselves, without waiting for the system to come to 

 rest. If / is the division which represents equilibrium, we may take 

 the value 



(35) 



I / 'X^ + X-, X,+ Xy\ 



which represents the mean position -I- -f -I of the 



2\ 2 2 / 



middles of two successive amplitudes. 



If this first approximation is insufficient, by writing that the three 

 successive deflections are in geometrical progression, and that 



from which follows 

 or 



(36) 



- 2X 



The division/ thus defined divides the oscillation into two parts, 

 which are as i = e~\ This point moreover has the property that 

 successive transits in opposite directions take place in equal intervals. 



This would not be the case with the perturbed motion described 

 above (684), in which disturbing forces come into play proportional 

 to the squares of the deflections. If / x be the division which corres- 

 ponds to equal times of oscillations on either side, the divisions / and 



* GAUSS. Restiltate aus den Beob. des Mag. Vereins. 1837. (Euvres, v., 

 P- 374- 



