OBSERVATION OF SWINGS. 267 



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886. OBSERVATION OF SWINGS. When we can control a phe- 

 nomenon under identical conditions, the experiment may be repeated 

 a great number of times alternately in the two directions, so as to 

 eliminate accidental errors of reading as well as displacements of 

 zero, and the mean of observations is taken. In this case the method 

 of multiplication enables us to still increase the angle of swing if we 

 make the discharges alternately each time the needle passes zero. 



The needle, being first started with a velocity o> by a first 

 positive discharge, reverts to its position of equilibrium after a time r 

 with the velocity - a> ^ / at this instant a negative discharge gives 

 it the velocity -w , which makes -G> O (I+^) J on i ts return to zero 

 it will have a fresh velocity + <*> , and so forth. The successive 

 deflections a lt a 2 , a 3 ---- being proportional to the velocities at the 

 various transits, we shall have 



The limiting elongation a^ corresponding to an established system 

 of inversions, and the initial elongation a, which will be produced by 

 a single discharge, are connected by the ratio 



The value of a defined by this equation, is expressed in divisions 

 of a scale, and we shall deduce the corresponding angle c^ from the 

 distance of the scale. 



This method has, moreover, the advantage of substituting a 

 permanent and regular regime for a single swing, which facilitates 

 readings. But, unless the initial swing is very feeble or the damping 

 considerable, we soon attain undue elongations ; for the value of a 

 increases without limit when A tends towards zero. 



We may by other methods effect a regular regime with deflections 

 of the same order as those of the first swing. 



887. The method of recoil was devised by Weber.* The needle 

 having been swung in the positive direction, it is allowed to acquire 



* W. WEBER. Abh. der Kbnigl. GeselL zu Gottingen, Vol. I., p. 349. 

 Resultate des Magn. Vereins, 1838, p. 98. 





