4L2 
PROFESSOR G. H. DARWIN ON FIGURES OF 
Hence, to the 7th order inclusive, we have 
V = 
47 to? a 
+ 
47 rA 3i ' = 6 [CL W 
V 
3 r 3c k = 2 \cj [2(7 —l)\r 
3 
lc + l 
+ + i&ua 2 
+ 
+ 
rVl Wh 
W 2 — i 8 2 w 4 
r~ 
■ • (78) 
Now the expression (72) for the radius-vector of the mass a to the same order of 
approximation gives us 
r 5a> 2 /iW' = 6 27 + 1 fay-2 
a + 16 tt V r 2 / \ c/ 27* - 2 \c) r* 
and a similar expression for R/A. 
To determine the inward force at the pole of the mass a, where it is nearest to the 
mass A, we must evaluate — dV/dr, and in the first term substitute the above 
expression for r, and in the remaining terms put r = a ; also at this pole wji' 3, = 1, 
and S hv± — 0. 
Then, differentiating (78), 
d V Area a? 
dr 3 r 2 
Ana /A\* k = 6 (aV~* 
k = 2 
{ 
3 (k + 1) 
2(7 — 1) 
+ & 
e> o 
— -fora 
3.3 
nord 2 | — 0 +2 
But at the pole 
47t« a 3 47ra 
3 r 2 
3 
4tt« M\3 * = <* 2k + 1 taV-z 
3 \c J k = 2 k — 1 \c) 
(79) 
Substituting this for the first term of (79), we have 
dV Ana , „ 0 
\rrrn f A \ 3 & — 6 
w(7 ,!,<* + *> 
But 
hence 
47ra 
43^4 3 + 13a 3 
12c 3 
3 
