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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)
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
— r I   J-i (rx) J"0 (ZI(K) J0 (lao&) ... JQ (Zna?) rfa, ...... (28)
— 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)
If in (29) we suppose r = l,
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,