384 
PROFESSOR G. H. DARWIN ON FIGURES OF 
We have now to find the potential at any point in space. 
The mass of the spheroid o is ^na z , and its potential is ^na^/r. 
The potential due to the departure from sphericity, represented by the term in 7q in 
the first of (17), is 
47rA 3 3 hi (aV fa\ i + l w ; 
3c~ 2A—2 \c) \r) r 1 ' .( 18 ) 
This is written in a form convenient for passing to the case of r 
be written in the form 
3hi c'wi 
/ (j \ 2i + 1 
- 
2 i -2r 2i + 1 ’ 
a. It may also 
(19) 
when it is in a suitable form for application of the transference formula (4). 
We shall now introduce two new symbols, namely, 
y = 
a 
Then (19) may be written 
r = (jj. 
a \ 3 3 7 ' 1 c'Wi 
( 20 ) 
2i — 2\ r ~' +1 ’ 
and, of course, the similar potential with the other origin is 
# 7ra 3 > — 
A \ 3 SffiP- 1 eWj 
2i —2 E~ i + 1 
( 21 ) 
The whole potential at any point of space consists of the potentials of the two 
spheres and of the inequalities on each. The potential of the inequalities of the sphere 
o may be written in the form (18), and of sphere 0 in the form (21). 
Thus the whole potential is 
-§7ra 3 
a Air A 3 3 hie faV /a'^ + l %o k 
r + 2k — 2 \c) \r) A 
(22—i.) 
+ 
4t rA 3 
3c 
c 47 ra 3 /A^ 3 
R + 3^ U 
3JTj P- 1 &Wj 
2 i — 2 iP + 1 
(22-ii.) 
The first line of (22) refers to origin o, the second to origin 0, and to this latter 
half the transference formula (4) must be applied. 
Now apply (2) to the first term of the second line, and (4) to one term of the series 
in the second term, and we have 
and 
Air A 3 c 
Air A 3 
‘A® fa\ k Wk 
3c R 
3c 
,!oW <*' 
47 ra 3 
(A' 
\ 3 3Hi P - r 
c ; Wi 
47r^4 3 
fa\ 3 35) P" 1 *=- 
3 
u 
/ 2i - 2 _ 
72 2i + i 
“ 3c 
\c) 2i — 2 k = o 
k + i ! 
a A io t 
