194 
SIR G. H. DARWIN ON THE 
k = / X ^ (l-fa 2 ) 1 ' 2 
a VI+ X/ (al)) 1 ' 3 
1,3 
we have 
Hence 
a _ k cos y _ 
a a sin /3 
Therefore 
Therefore 
These enable us to differentiate, with respect to a, t), functions expressed in terms 
of a, 6, c ; the parameters r, f s always occur explicitly. 
The equation of condition for the parameter r is 
or or 
( x ' 
1/3 a 2/3 
6 
_ * 
cos 
P 
J * \ 
1/3 ju 2 /: 
c 
k 
_/ X V 
Vl+X/ 
) lj 1/3 
? 
a 
a 
sin 
P 
u+xy 
a i: 
a 
a sin /3 
Vl + X/ 
, da 
2— - 
_db 
, 3 
db . 
-*+2 
db 
r ° 1 
CO 
_ _ da _ 
dtr 
a 
a 
1) 
6 
a 
b ’ 
c 
a 
b ’ 
3a 
= 2a 
0 
b~ 
—c 
0 
or 0 
31) — 
0 
Cl — 
+ 26--c 
0 
3a 
3 a 
06 
0C ' 
01) 
0a 
06 
0c ’ * 
On differentiating (22) we have 
1 2 3^ _ 4, 
XoV 
In order to differentiate V we must take separately its several portions as defined 
in (21). 
Now 
l(eE h = (I 
X 
d 
f r \ 
(vv) = (|-7rpa 3 ) 2 ——| - £ t //$ / Wiil 1 ) Ci*(dv) — <Qf ^j (f>2, s even) 
(all harm. 
, 9 ^_ 2 V (//) 2 S7 cos 2 y 
2 r 4 2i+ 1 sin 2 /3L 
5(3 + k 2 )£ 2 5(5 + 2k 2 +k 4 )& 4 
14k 2 r 2 
56k 1 
v~( HF)= symmetrical expression for larger ellipsoid. 
P ill 
(1+X) 2 l2.5r 4 L V 
1 
2 /J_ 
K 2 
+ 1 
+ 
+ 
+ 
+ 
2 3 .7r 6 
3 
2 2 .5r 6 
1 
il (S + l + 3 ) + ^‘(l + l +8 
" 2 / i 2 + ^ 2 + iT 2 + 3 
*"(4 + ?a + 3 + 5)+^f#. + li + ^ + 5 
2 1 . 3 r 8 L \k 6 k 4 k' 
9 
2 4 .5r 8 
P^(-4-. + i + 
K K 4 K 2 
2,2 1 
+ 772 + “ 2+ 5 
K 2 K 4 K 4 k 2 K 2 K 2 k 
1 . 2 
WFli + — + -2W2 + 1 + G2 + 0 
lv K K K Iv K K 
