1919]
ON RANDOM VIBRATIONS AND RANDOM FLIGHTS
605
or in terms of r, 6,
J^
Trn
(4)
Thus all phases are equally probable, and the chance that the resultant amplitude lies between r and r + dr is
-e-r2'nrdr..................................(1)
n ^ '
This is the same as was before stated, but at present the conditions are limited to a distribution of precisely \n components along x and a like number along y. It concerns us to remove this restriction, and to show that the result is the same when the distribution is perfectly arbitrary in respect to all four directions.
For this purpose let us suppose that ^n + m are distributed along' ±x, and $n — m along ± y, and inquire how far the result is influenced by the. value of TO. The chance of the representative point lying in rdrdQ is now expressed by
_______________
TT ^(n? — 4m2)
Since r is of order \/n, and m/n is small, the exponential containing d may be expanded. Retaining the first four terms, we have on integration with respect to d,
2r^T — *-**(*-*#) fl
V (n2 - 4ro2) [
as the chance of the amplitude lying between r and r + dr. Now if the distribution be entirely at random along the four directions, all the values of TO of which there is a finite probability are of order not higher than \?n, n being treated as infinite. But if m is of this order, the above expression becomes the same as if TO were zero; and thus it makes no difference whether, the number of components along ± no and along + y are limited to be equal, or not. The previous result is accordingly applicable to a thoroughly arbitrary, distribution along the four rectangular directions.
The next point to notice is that the result is symmetrical and independent of the directions of the rectangular axes, from which we may conclude that it has a still higher generality. If a total of n components, to be distributed along one set of rectangular axes, be divided into any number of large groups,, it makes no difference whether we first obtain the probabilities of various, resultants of the groups separately and afterwards of the final resultants, or whether we regard the whole n as one group. But the probability in each group is the same, notwithstanding a change in the system of rectangular axes; so that the probabilities of various resultants are unaltered, whether
[* A correction of sign here made, viz. "-+2mr2cos20" for " - 2mr* cos 20," applies also to* Vol. i. p. 494, lines 10 and 12. W. F. S.] " : :part removed, with nitric acid, so as to allow the transmission colour to be observed. But a much superior effect has been obtained by Dr Eltringham, using eau de javelle (hypochlorite)*.