606 ON THE PROBLEM OF RANDOM VIBRATIONS, AND [441
we suppose the whole number of components restricted to one set of rectangular axes or divided in any manner between any number of sets of axes. This last state of things is equivalent to no restriction at all ; and we conclude that if n unit vibrations of equal pitch and of thoroughly arbitrary phases be compounded, then when n is very great the probability of various resultant amplitudes is given by (1).
If the amplitude of each component be I, instead of unity, as we have hitherto supposed for brevity, the probability of a resultant amplitude betweeen r and r + dr is
.................... ; ............. (5)
nl*
In Theory of Sound, 2nd edition, § 42 a (1894), I indicated another method depending upon a transition from an equation in finite differences to a partial differential equation and the use of a Fourier solution, This method has the advantage of bringing out an important analogy between the present problems and those of gaseous diffusion, but the demonstration, though somewhat improved later*, was incomplete, especially in respect to the determination of a constant multiplier. At the present time it is hardly worth while to pursue it further, in view of the important improvements effected by Kluyver and Pearson. The latter was interested in the " Problem of the .Random Walk/' which he thus formulated : — " A man starts from a point 0 and walks I yards in a straight line ; he then turns through any angle whatever and walks another I yards in a second straight line. He repeats this process n times. I require the probability that after these n stretches he -is at a distance between r and r + dr from his starting point 0.
" The problem is one of considerable interest, but I have only succeeded in obtaining an integrated solution for two stretches. I think, however, that a solution ought to be found, if only in the form of a series in powers of 1/w, when n is large f." In response, I pointed out that this question is mathematically identical with that of the unit vibrations with phases at random, of which I had already given the solution for the case of n infinite J, the identity depending of course upon the vector character of the components.
In the present paper I propose to consider the question further with extension to three dimensions, and with a comparison of results for one, two, and three dimensions §. The last case has no application to random vibrations but only to random flights.
* Phil. Mag. Vol. XLVII. p. 246 (1899) ; Scientific Papers, Vol. iv. p. 370.
t Nature, Vol. LXXII. p. 294 (1905).
J Nature, Yol. LXXII. p. 318 (1905) ; Scientific Papers, Vol. v. p. 256.
§ It will be understood that we have nothing here to do with the direction in which the vibrations take place, or are supposed to take place. If that is variable, there must first be a resolution in fixed directions, and it is only after this operation that our present problems arise. from both sides (1917).ds; and since the range of relative index is then much restricted, the