GAUSS'S METHOD OF REDUCTION. 45 



The first member of this equation which represents the relative error 

 made in the determination of A has the least value when the product 



bl. - , considered as a function of b, is a maximum that is to say, 

 b 



when 



/ a I 



/ .,, or *- r 



With the series of observations of Gauss, the logarithmic decrement 

 A is determined according to the case, either by amplitudes of the 

 same series, 



or by corresponding amplitudes of two consecutive series separated 

 by n i oscillations. 



n a. 2 



and finally and more simply, by the mean amplitudes a v a', a", of the 

 different series. 



By calculating A, from the successive oscillations of the same 

 series, or from corresponding oscillations of two successive series, we 

 retain the advantage of verifying whether the amplitudes do vary in 

 geometrical progression. 



698. We shall reproduce here the example given by Gauss 

 himself, with some change in the arrangement of the columns, and 

 suppressing the last column of decimals which may be considered 

 illusory. This observation is on the oscillations of a magnet of 

 twenty-five pounds under the influence of the earth alone. 



The times observed are those at which the line 1,000 of the scale 

 passes over the cross-wires. The values of the position of zero 

 deduced from formula (35) have been added to the table. These 

 values do not enter into the calculation, but it is necessary to be 

 certain that the zero has not varied sensibly in the course of the same 

 series, and therefore in the interval of two consecutive passages, for a 

 calculation of the time of elongation is only exact as long as this 

 latter condition is fulfilled. 



