# Full text of "Scientific Papers - Vi"

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```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 v°v"'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 n«2    /  »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.
```