1919] RANDOM FLIGHTS IN ONE, TWO, OR THREE DIMENSIONS 613 Taking first the integration with respect to 0,, we have by (25) <wid thus P2(r- llt lz) = r \ dxJ, (rx) J,(l,x} J,(l,x) ............. (27) JO The method can be extended to any number (n) of stretches. Beginning with the integration with respect to 6n^ in (26), we have as before If r (%* ft0 ^~ ddn--,. = Q- d0n_! dxJt (rx) J"0 (snx) z'7r J "IT JQ ./O = rl dx Jx (roe) J0 (lnx) J0 (SH_Z x). J o The next integration gives i rr r°° (27r? J J d^n-zdOn^ = r I /j. (rx) J0 (lnx) J0 (In^x) J0 (Sn-sŤ) dx, and so on. Finally rco r I J-i (rx) J"0 (ZI(K) J0 (lao&) ... JQ (Zna?) rfa, ...... (28) Jo the expression for Pn discovered by Kluyver. It will be observed that (28) is symmetrical with respect to the Z's; the order in which they are taken is immaterial. When all the I' 8 are equal, Pn(r; 0 = ^r^(ra){J0(te)}M^ ................... (29) Jo If in (29) we suppose r = l, >W1' .(30) so that after n equal components have been combined the chance that the resultant shall be less than one of the components is lj(n+ 1), an interesting result due to Kluyver. The same author notices some of the discontinuities which present themselves, but it will be more convenient to consider this in a modified form of the problem. The modification consists in dealing, not with the .chance of a resultant less than r, but with the chance that it lies between r and r + dr. It mayi). * Similar reasoning shows that if D0 (g) represent a symmetrical purely divergent wave,