REDUCTION OF OSCILLATIONS. 37 



If the observations are made to within of a second, we shall have 



10 



for a quarter of an hour an approximation of about . 



If there is no interest in verifying the exactitude of the cor- 

 rection of the deflections separately for different series, the reduction 

 may be made more rapidly. Let / 15 / 2 . . . . p b be the numbers of 

 the infinitesimally small vibrations of times r which are made during 



the intervals ^ - / , f t2 -f v / 3 - 1 2 / 5 - / 4 of the twenty vibrations 



which form the successive series ; we have 



P being the total number of infinitely small vibrations which 

 would correspond to the whole time / 5 -/ of the five series, it 

 follows that 



More generally, if during the time we have observed m series 

 of n vibrations that is, a total of nm = N vibrations and that the 

 corrections of the mean amplitudes are ft, ft /? m , we have 



B 

 and T -- . 



m J P 



692. When the deflections are so small that the value of /? 



a 2 

 may be reduced to its first term , and that the amplitudes are in 



geometric progression, the correction may be simply made by means 

 of the logarithmic decrement. The successive times of n vibrations 

 will be 



T ( 1 4- B I T/T -I- 6 *~2A| T (j 4- fi /> 2( 1)A\ 



r i * T [~ J \i ) v r^l* yj**'* I * ! /-'i*- I , 



