Vibrations of the same Pitch and of arbitrary Phase. 77 



Now, if the distribution be entirely at random, all the values 

 of m of which there is a finite probability are of order not 

 higher than v" w, n being treated as infinite. But if m be of 

 this order, the above expression is the same as if m were zero, 

 and thus it makes no difference whether the numbers of com- 

 ponents along ±a and along ±y are limited to be equal or 

 not. The previous result, viz. 



2 _— 



- e .» r dr. 



n ' 



is accordingly applicable to a thoroughly arbitrary distribution 

 among the four rectangular directions. 



The next point to notice is that the result is symmetrical, 

 and independent of the direction of the axes, so long as they 

 are rectangular, from which we may conclude that it has a 

 still higher generality. If a total of n components, to be dis- 

 tributed along one set of rectangular axes, be divided into any 

 number of groups, it makes no difference whether we first 

 obtain the probabilities of various resultants of the groups 

 separately and afterwards of the final resultant, or whether we 

 regard the whole n as one group. But the resultant of each 

 group is the same, notwithstanding a change in the system of 

 rectangular axes; so that the probabilities of various resultants 

 are unaltered, whether we suppose the whole number of com- 

 ponents restricted to one set of rectangular axes or divided in 

 any manner between any number of sets of axes. This last 

 state of things, however, is equivalent to no restriction at all; 

 and we thus arrive at the important conclusion that, if n unit 

 vibrations of equal pitch and of arbitrary phases be com- 

 pounded, the probability of a resultant intermediate in ampli- 

 tude between r and r + dr is 



2 - r - , 



- e n r dr. 



n ' 



a similar result applying, of course, in the case of any other 

 vector quantities. 



The probability of a resultant of amplitude less than r is 



I 



2 -.- _- 



-e~ n rdr=l— e « 

 n 



or, which is the same thing, the probability of a resultant 

 greater than r is 



e »« 



The following table gives the probabilities of intensities less 



