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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.
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
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.
392opose 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.