(nI) pA B C
AJ B (I) (J) s=
dA h C _D + 1A )nC)X 8
SBC, D (I) J) 2 BC, DE o(I)(J) X s 0
+0(X3/4), (4-103)
where ABc is expanded around A = 0, and its expansion coefficients iAB, D and
0
iAC DE are computed via Eqs. (4-93) and (4-95) along with (4-91) and (4-92).
The first order approximation for XA(s, A1, A2, A3) near the initial point A = 0 is
axA axA
XA(s, A1, 2, A3) s + A' + [(so )2/1R]
0 0
s ~,)I + O[(s, )2/R], (4-104)
where A, A(1) n(2), n (3) are the orthonormal basis vectors evaluated at the
location of the particle, and the summation is assumed over the repeated index
I = 1, 2, 3 (hereafter, we omit the summation sign and assume the summation
convention for the up-and-down repeated spatial indices I, J, K, L,... = 1, 2, 3).
Substituting this into Eqs. (4-101)-(4-103), we obtain the quadratic approximations
for the integrands
( IU \ A j%CjDS/ t1 4 OiiCii
= X -0 BC, D C, DE 0o 0 0 0
+O(S3/R4), (4-105)
((i) A iB 1CfD t A iB iCDfE
as AO BC, D o(lJ) 0o 2 BC,DE o 0 (J)o0 U0
+O(S3/R4), (4-106)
(a ) ^A B (Ds + D K)
Oa j BC, D 'o(I) o(J) o S (K)
S0 (s +A D K) (s + o(L)A)
O sC,DE () (J) 0 (K) AK 0 (L) )
[ 3/]. (4 7
+o[(s, )3 /4]. (4-107)