FEB. 19, 1923 HUMPHREYS: THE MURMUR OF THE FOREST 61 
when all the vibrations are confined to two exactly opposite phases, 
+ and —. Now, the chance that all the n vibrations will have the 
same phase, + say, is (})" and the expectation of intensity correspond- 
ing to this condition (4)"n*. Similarly, when one of the vibrations 
has the negative phase and the n—1 others the positive phase, the 
expectation is ($)"n(n —2)?; and the whole or actual expectation 
yn jb +n(n—2)2 + MRD 
The sum of the n + 1 terms of this series is n, as may be indicated 
by a few numerical substitutions; or proved, as Lord Rayleigh (I. c.) 
has shown, by expanding the expression 
(n—4)? + .... (A) 
(e* + 6%)” 
into the two equivalent series 
2° (1 + 4027+ ...... a (Maclaurin) 
and 
tae SOY ate jie ae (Binomial) 
developing the exponentials into series of algebraic terms, and, 
finally, assembling and equating the coefficients of x? in the two 
equivalent series, and solving for n. The value of n thus found is 
identical with the expression (A). 
That is, on the average, the intensity of the resultant of n sounds 
of unit amplitude, but confined, in random numbers, to two opposite 
phases, is always n, whatever its numerical value. 
If, instead of the numbers of sounds in either of two opposite phases 
being random, the phases are random, the result, as Rayleigh has 
shown, is the same. 
It should be noted that n is only the mean intensity of a possible 
range from 0 to n?, and not the continuous intensity. But when the 
changes are rapid the fluctuations from the mean are correspondingly 
inconspicuous. 
From the above two laws, namely, (1) that the pitch of a composite 
note is the approximate average of those of its constituents, and (2) 
that the mean intensity is the sum of the individual intensities, it 
appears (a) that the pitch of the aeolian murmur of a forest is essen- 
tially that of its average twig, or needle, if the forest be of pine, and 
(b) that though the note of the twig may be inaudible, even at close 
range, the forest often may be heard miles away. 
