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.