78 On the Resultant of a large Number of Vibrations. 



than the fractions of n named in the first column. For ex- 

 ample, the probability of intensity less than n is *6321. 



•05 



•0488 



•80 



•5506 



•10 



•0952 



1-00 



•6321 



•20 



•1813 



1-50 



•7768 



•40 



•3296 



2-00 



•8647 



•60 



•4512 



3-00 



•9502 



It will be seen that, however great n may be, there is a rea- 

 sonable chance of considerable relative fluctuations of intensity 

 in consecutive trials. 



The average intensity, expressed by 



»00 O j-2 



i 



rdr 



is 



as we have seen already, equal to n. 



If the amplitude of each component be a, instead of unity, 

 as we have hitherto supposed for brevity, the probability of a 

 resultant amplitude between r and r + air ' 



2 



is 



n« 



no? f dr. 



The result is therefore in all respects the same as if, for ex- 

 ample, the amplitude of the components had been -J a and their 

 number equal to 4n. From this we see that the law is not 

 altered, even if the components have different amplitudes, 

 provided always that the whole number of each kind is very 

 great ; so that if there be n components of amplitude a. n' of 

 amplitude (3, and so on, the probability of a resultant between 

 r and r + dr is 



2 r 2 



— — - e n«2+„^2+. . . r ^ r% 



not? + n'fi 2 + . . . 



The conclusion that the resultant of a large number of in- 

 dependent sounds is pi-actically, and to a considerable extent, 

 uncertain may appear paradoxical ; but its truth, I imagine, 

 cannot be disputed. Perhaps even the appearance of paradox 

 will be removed if we remember that with two sounds of equal 

 intensity the degree of uncertainty is far greater, as is evi- 

 denced in the familiar experiment with tuning-forks in ap- 

 proximate unison. That the beats should not be altogether 

 obliterated by a multiplication of sources can hardly be thought 

 surprising. 



June 1880. 



