346 Lord Rayleigh : Problem of Random Vibrations. 



So far as the principal term in (70) is concerned, the 

 maximum value occurs when ?*//=2. 



It will be seen that the agreement of the two formulae is 

 in fact very good, so long as rjl does not much exceed */n. 

 As the maximum value of rjl for which the true result 

 ■differs from zero, is approached, the agreement necessarily 

 falls off. Beyond r/l=n, when the true value is zero, (70) 

 yields finite, though small, values. 



Terling Place, Witham, 

 January 24th, 1919. 



P.S. March 3rd. 



In (45) we have the expression for the probability of a 

 resultant (r) when a large number (n) of isoperiodic vibra- 

 tions are combined, whose representative points are distri- 

 buted at random along the circumference of a circle of 

 radius /, so that the component amplitudes are all equal. 

 It is of interest to extend the investigation to cover the case 

 of a number of groups in which the amplitudes are different, 

 say a group of p l components of amplitude Z 1? a group 

 containing p 2 of amplitude l 2 , and so on to any number of 

 groups, but always under the restriction that every p is 

 very large. The total number (%p) may still be denoted 

 by n. The result will be applied to a case where the number 

 of groups is infinite, the representative points of the com- 

 ponents being distributed at random over the area of a circle 

 of radius L. We start from (31), now taking the form 



foe 



The derivation of the limiting form proceeds as before, 

 where only one I was considered. Writing 5 1 = ^ j p 1 Z 1 2 , 

 ■s 2 = ^p 2 l 2 2 f & c - 5 we have 



log[{J (^)}^{J (U0K 2 ] 



~^->-f>e)-fH')' 



-and thus 



