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.