1919] RANDOM FLIGHTS IN ONE, TWO, OR THREE DIMENSIONS 611
From (22) the successive forms are determined graphically. For values of n higher than 7 an analytical expression proceeding by powers of l/n is available, and will be further referred to later.
A remarkable advance in the theory of random vibrations and of flights in two dimensions, when the number (n) is finite, is due to J. C. Kluyver*, who has discovered an expression for the probability of various resultants in the form of a definite integral involving Bessel's functions. His exposition is rather concise, and I .think I shall be doing a service in reproducing it with some developments and slight changes of notation. It depends upon the use of a discontinuous integral evaluated by Weber, viz.
r=o
Jl (bx) J0 (ax) dx = u (say). Jo
To examine this we substitute from
TT . J^ (bx) =2 cos B sin (bx cos 0) d0\t
Jo
and take first the integration with respect to x. We havej
dx sin (bos cos 0) J0 (ass) = 0, if a2 > 62 cos2 0, o
or = (b2 cos2 6 — a2) ~*, if &2 cos2 B > a2.
Thus, .if a? > fr, u = 0. If 62 > a2,
2 f d0cos0 2 . , &sin<9
U - - I -TV;-----r-0------=r = —f Sin
osj a — a ) TTO
The lower limit for B is 0, and the upper limit is given by cos2 0 = a?jb*. Hence u — 1/b, and thus
b r Jl(bx)J,(ax)dx=l, (&2>a2)\................^
or = 0, (a2 > ft2) J
A second lemma required is included in Neumann's theorem, and may be very simply arrived at. In Fig. 1, Q and E being fixed points, the function at F denoted by
J0(g), or J0V(e2+/2-2e/cos<?)) is a potential satisfying everywhere the equation V2 + l=0, and accordingly may be expanded Q round G in the Fourier series '
A0J0 (e) + A,J, (e) cos G + Az J, (e) cos 2(7 + ..., Pig' L
the coefficients A being independent of e and G. Thus 1
* Koninklijke Akademie van Wetenachappen te Amsterdam, Verslag van de gewone verga-deringen der Wis-en-Natuurkundige Afdeeling, Deel xiv, 1st Gedeelfce, 30 September, 1905, pp. 825-334.
f Gray and Matkews, BesseVi Functions, p. 18, equation (46). f G. and M., p. 73.
39—2opose 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.