Vibrations of the same Pitch and of arbitrary Phase. 75 



of n vibrations, which are at random positive or negative, the 

 number of positive vibrations lies between 



is, when n is great, 



4-{ T e- t2 dt, 



where T=/' s /(2ft), and r must not surpass s/n in order of 

 magnitude. In the extreme cases the amplitude is ±2r v / (i^)j 

 and the intensity is 2-rn. Thus, if we put t= J, we see that 

 the chance of intensity less than J« is 



-4-fV /2 (7f = -5205; 



so that however great n may be. there is alwavs more than an 

 even chance that the intensity will be less than \n. This, of 

 course, is inconsistent with any such tendency to close upon 

 the value n as Yerdet supposes. 



From the tables of the definite integral, given in De Mor- 

 gan's * Differential Calculus/ p. 657, we may find the proba- 

 bilities of intensities less than any assigned values. The pro- 

 biiity of intensity less than £n is '2764. 



Again, the chance that in a series n the number of positive 

 vibrations lies between 



|»+7V(i*) and \n + {T + hr)s/{\n) 

 is 



which expresses accordingly the chance of a positive amplitude 

 lving between 



2tV(|w) and 2(T + oV) N /(i'0- 



Let these limits be called x and x + Sx. so that r = x ^(2n); 

 then the chance of amplitude between x and x+8x is 



e 2/i ox. 



s/{2tth) 



The expectation of intensity is expressed by 



i r +x _' 2 



— r— rl e 2"x 2 dx=n 



as before. 



It will be convenient in what follows to consider the vibra- 

 tions to be represented by lines (of unit length) drawn from a 

 fixed point 0, the intersection of rectangular axes Ox and Oy. 



G 2 



