NOTE ON THE EXPANSION OF THE TRUE ANOMALY. 
140 
[191 
considered is therefore that of s = r, in which case there is a term containing e r . We 
thus obtain from \~~ r $ re the term 
1 
2 
r r e r 
2'. 1.2.3 ...r' 
In the next place a term of the form Ve s (A — A _1 ) s is at least of the order e® 
if s > r, or the terms to be considered are those for which s = or < r. But in such 
term the only part of the order e r is 
(-)* v-v, 
or, since neglecting higher powers of e we have A = this is 
(_) s 2~ r+s e r , 
and the set of terms arising from 
A r 
is 
Oî* ^ ~\ '*) * * * ^ T f 
+ 
1 1.2 1.2...(r-1) 2 1.2...rj ' 
the last term being divided by 2 because arising from a term independent of A. 
Hence the first term of A r is 
e r , r I r 2 i r r ] 
2 r I d - ^ d - ^ 2 * * * 1 2 r I ’ 
a result which it may be remarked is contained in the general formula given in 
Hansen’s Memoir “Entwickelung des Products u.s.w.,” Leipzig Trans., t. II. p. 277 (1853). 
The preceding expression is 
e r c r 
2 r T (r 
1 [ 
MÖ)J f xr °~" dx ’ 
and to find its value when r is large, we have 
J x r c~ x dx = j (y + r) r e~ v ~ r dy = r r c~ r j f 1 + e~ y dy 
. r ^-3/+'log(l+l) 
J 0 
= r r c~ r J C 
.00 y - , v 
— r >Q-r 
"27 + 3^-& c - 
dy 
dy 
/»oo , 0 \ 
-W. ( 1+ ^ + -)«^ 
\/-2r f 
J 0 
— rc 
1 + —^ z 3 + ...) e~ z * dz, 
3 Vr