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1919]         RANDOM FLIGHTS IN  ONE, TWO, OR THREE  DIMENSIONS              621
This argument must appear, very roundabout, if the object were merely to obtain the result for n = 2. The advantage is that it admits of easy extension to the general value of n. To this end we take the last stretch ln and the immediately preceding radius sn^, in place of lz and ^ respectively, and then repeat the operation with ln-i, sn-Z) and so on, until we reach 1% and s1 (= /a). The result is evidently
7   sin roc— rx cos roc sin l^x sin I2x    sm.lnx etas                    -                        ___...__,
...... (5V)
or if we suppose, as for the future we shall do, that the I's are all equal,
„  ,     7.     2 f°° 7   sin rx — roc cos rx /sin loc\n
,„„. (58)
^       '
'                  7T    o     •                                  OS
This is the chance that the resultant is less than r.    For the chance that the resultant lies between r and r + dr, we have, as the coefficient of dr,   -
rJP        9r   f00   clr
~r=^n      :~-Bin^sin»te ................... (59)
dr      7rlnJQ  %n x                                                          x    }
Let us now consider the particular case of n = 3, -when dP,     2r    °° dx .         .
w//             /< (/   j ()      w
In this we have                                                                 .
sin rx sii\3lx — %(3 cos (r — 1) x — 3 cos (r + l)x- cos (r — 31) x •+ cos (r + 3Z) *•}.
/-°° ^
And          — {cos (r - Z) x — cos (r -f 1} x}
Jo   x
and in like manner for the second pair of cosines.
Thus                ~? = ^{2r-3 \r~l\ + \r-Bl\} ..................(61)
expresses the complete solution.    When                          _.
It will be observed that dPa/dr is itself continuous ; but the next derivative changes suddenly at r = I and r = 3Z from one finite value to another.
Next take n =4.    From (59)
- = —n I    —r sin rx sin4 Ix,
TTt4 J o    Xsto deal with their cosines, of which all values are to be regarded