184 
SIR G. H. DARWIN ON THE 
The last term, being independent of x, y , 2 , has no present interest. Then, collecting 
results from ( 10 ), ( 11 ), and ( 12 ), the whole potential, as far as material, is 
a M 
2 k 
'_ 2 _ 
3A 
3©i 
x 
*>„*-!/**) 
\cu 2 k 2 r | 2 
k) ~ ( 1 +X) M J k 
/\ 1 _|_ nr~ i/k ~ F 
3\ 
\ ^ 3(3 + k 2 )F , 5(5 + 2k 2 + k 4 )H 
14k 2 ~ r 2 ' 
56k 4 
* 
2T 
a-W-i) 
& 1 +*'° 3 IF 
3\ 
1 + 
3 _(3k 2 +1)F 5 (5k 4 + 2k 2 + l) & 4 
14k 2 '» i2 Kfi- 4 ^.4 
56k 4 
oi 2 F 
SM 
(V-l) 
kW 
3i- 
2v 2 k s 
3\ ? 
*3 
'l + 3(1 + F) F 5 (3 + 2k 2 +8 k 4 ) IP 
7k 2 r 2 56k 4 r 4 " 
eo 2 F 
3ilT 
p;-.' 
The condition that the figure of equilibrium should be the ellipsoid of reference is 
that tins potential when equated to a constant should reproduce the equation to the 
ellipsoid. I he coefficient of 2 must therefore vanish, and the three coefficients written 
inside { } must be equal to one another. These conditions give the angular velocity 
and equations for determining the figure, but as the subject will be reconsidered from 
a different point of view hereafter, I do not pursue the investigation here. 
At present it need only be noted that the coefficient of 2 vanishes, and that the 
three coefficients are equal to one another. It is clear then that the potential U of 
the system, as rendered statical, may be written 
jj — _3jTJ 1 3 (3 + k 2 ) F ( 5 (5 + 2k 2 + k 4 ) k 
2k 
3X 
14 k 2 
56k 4 
x 
+ 
yr 
F(r„«-1/^) ' F(V-l) + iV 
Now gravity g at the surface of the ellipsoid is — dU/dn , where n is the outward 
normal to the ellipsoid. 
Hence 
9 = ~ 
Now 
px 8 U py dU , pz dU 
Lf (v 2 -1 /k 2 ) a.x k 2 (v 2 - 1 ) a y k\ 2 82 _ 
1 
x 
t? v (*„ 2 - i/*y + ** ( v ‘- 1 y T hw ’ 
and in our alternative notation 
„ 1 COS 2 y 
yr 
+ 
D, — -» — 
k 2 sin 2 /3' 
Therefore 
9 = 
3 M 
pk 
A 1 | F cos 2 y 
1 3Ar a sin 2 f3 [_ 
, 3 (3 + k 2 ) F , 5(5 + 2k 2 +k 4 )F , 
L ' 14k 2 r 2+ ^ + 
56k 4 
(13). 
