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Full text of "Scientific Papers - Vi"

1919]         RANDOM FLIGHTS  IN  ONE,  TWO,  OR THREE DIMENSIONS             '617
The evaluation of the principal term depends upon a formula due, I think, to Weber*, viz.
(raj) e-^asdx = ~ <r*'& , ... ................ (39)
~
u =
o making -f-
dr    J0 vv"'v              2p'J0 - J_ C0  -p'Vf r//,
Hence                                   w = <7e-W.
To determine G we have merely to make r = 0.    Thus
;by which (39) is established.
Unless las is small, the factor {J0(los)}n in (32) diminishes rapidly aa ?i increases, inasmuch as J0 (Ix) is less than unity for any finite las. Thus when n is very great, the important part of the range of integration corresponds to a small las.
Writing 5 for ^nP, we have
16w2
16h9
"   ^-*
i,                                       /-T/TVI^                 1 n2    /  i "          "   ^-*               O   tO
so that        _         {J. (to)}- = r+* (1 - ^ -:?_ +
/oo                                           /            (j2T4         n3^,6            o4.8   \
making    2^ (,f) ./o .*/, (r.) r** (l -  - -^ + ^ .  . . .(40)
Calling the four integrals on the right J1; 72, 73, and J4, we have by (39)
xye^^^-e-7-^, ..................... (41)

}3 /yar       cs    /7s
r - JL i - __    ^ 
3                        ~"
* Gray and MathewSj loc. cit. jx 77,
t I apprehend that there can be no difficulty here as to the differentiation, the situation being dominated by the exponential- factor.                       ."...:.,_                .         . . 'inities are excluded if n > 9, and so on.