should be also replaced by ( ') JtA//f (r + J2). Then,
J k-p
(- Q)k-P __') 7 _2)
k-p
= ACp ( k-p-i ( )Ai k-p-i/Z--i (4-163)
i=0
S (sin 0)' (cos 4) Ak --i /P k--i + O[(x Xo)k-p+2], (4-164)
where a binomial expansion over the index i = 0, k p is assumed with the
coefficient ckpi 1/Rk-p-i in Eq. (4-163), and in Eq. (4-164) ( 0')' is replaced
by [sin(O o')]1 + O[(4 #')i+2] the term O[(x xo)k-p+2] at the end results
from this O[(0 ,')i+2], then the coordinates are rotated using the definition of
new angles by Eq. (4-151). Also, by Eq. (4-151) again
( 5 (sin O)P (sin )"p + O[(x- xo)p+2]. (4-165)
Using Eqs. (4-164) and (4-165), the behavior of Qa[c-2] in Eq. (4-162) now looks
like
Qa[e-2] 3A1-p-i (sin Q)p+i (sin ))P (cos )i Z, (4-166)
where s = p+i for a = t, r and s = p+i+1 for a = 0, and any contributions from
O[(x- xo)k-p+2] in Eq. (4-164) and from O[(x- xo)p+2] in Eq. (4-165) have been
disregarded: by putting these pieces into Eq. (4-162) we simply obtain eo terms or
0(C2), which would correspond to C,-terms or (0-4) in Eq. (4-33), and later in
this Section it is proved that they ahv--, vanish. Qa[c-2] in Eq. (4-166) then can be
categorized into the following cases:
(i) i 1
The integrand for the integration-averaging process over K) is F() -
(sin )"P (cos )i = (sin )" (cos )), and it has the properties that F(4) + r)