424 
PROFESSOR G. H. DARWIN ON FIGURES OF 
whence, with all the terms, and remembering that (a/c ) 2 = 7 , (R/c ) 2 = r, 
e, - 
--*[(' 
47t \ 2 R 3 a 3 " 
3 
R\ 3 * = ” 27,• 4- 1 7 
j-i 
2 
k = 2 2/c 
2 & - 1 
M* 2 +- 
1c + 2! 
2 . 1 ; —2 \ 
pr 
(ii.) 
The formula for JS 7 may be written down by symmetry. 
2 nd. e 3 , the exhaustion of mutual energy of the layer on itself, is half the potential of the layer at 
any element, multiplied by tbe mass of the element, and integrated over the whole sphere. 
The potential of the layer is 
3 
3c tio 2k-2 
„ Wk 
nk - r i 
Then, at an element of the layer r = a, and taking a typical term only, w r e have 
whence 
1 4,7R 3 3(27.: + 1) 
** _ 2 ' 3c ‘ (2 k - 2 )~ 
2k- 
A\ 3 
nk 
w k\ , 
3 + P* 
v Wjc + 2 
e 2 
2 , fc + 2! 
n* 2 + 
2.7c — 2! 
pr 
(iii.) 
The formula for F7 3 may be written down by symmetry. 
The addition of to e 3 , and of F7 1 to F7 3 , simplifies these expressions by cutting out the factor 
immediately following the in either, and replacing it by unity. 
3rd. e 3 is the loss of energy due to raising the layer on a in presence of the mean sphere R. We 
multiply the potential of the sphere R by the mass of the element on n, and integrate throughout angular 
space. 
The potential of the sphere R, when transferred to the origin 0 , is 
(A* (A*fto m 
3c k = 0 \c) \a) r k 
Then, at an element of the layer r = a and taking a typical term, 
4ttR3 2 A: + 1 /R \ 3 /a\~ k ~ 
e> — 
3c 2 /i - 2 V c j \c j 
whence 
Co = 
477 -\ 2 R 3 a 3 
3 
MJ 
d\ 3i '=°° 7^-1 
C k = 2 
L- = 2 7l — 1 
n*. 
(iv.) 
The expression for E s may be written down by symmetry. On collecting results from (ii.), (iii.), 
and (iv.), we have 
e l + e 2 + e 3 ~ 
4tt \ 2 R % 3 
3 
• 3 . 
2 
R\3 k = co r .k- 1 
' V _ 
1 = 8 7,- - 1 
91 
nk — ¥<-r — 
4.7,-- 2! 
pr 
(v.) 
and a similar expression for + h J 3 + JE7 3 . 
4th. (F7e ) 4 is the loss of energy of one layer in the presence of the other. We take the potential of 
the layer on R, multiply it by the mass of an element on a, and integrate. 
The potential of the layer on A when transferred to the origin 0 , as in (22—ii.), is 
47tR 3 9 /a\ 3 t ~ 00 p + i\ F 1 /<i\ k /rV'Wk 
~B 7 He/ U = 2 1 = 2 L*! H i^l JVi [cj W r* 
fe + t! e!z1 P 
*- 2 !fc + 2 !t-l ‘ 
Introducing this into the integral, only taking a typical term, and neglecting those terms in the integral 
which must vanish, we get 
