226 
SIR G. H. DARWIN ON THE 
In this way I find the following expressions for the semi-axes in series proceeding 
by powers of <f or sin 2 y :—- 
_ 
9 
a 
X \ 2/3 
l + x 
\ , 7 + X 2 , 524 + 223X +14X 2 4 
1 + —/+ OB „ /, . — 9 
3 (4 + X)' 
3 2 .7 (4 + X) 2 
120926 + 86748X+ 19428X 2 + 686X 3 6 
+ - x - . . cf. 
3 4 .7 2 (4 + X) 3 
a 
a 2 
X \ 2/3 
1+X 
5 + 2X 
I_ ^ i 2 _64 + 8X + 7X“ 
3 (4 + X)^ 3 2 .7 (4 + X) 
2 
2 sr 
11122 + 2460X —1851X 2 + 196X 3 6 
3 4 .7 2 (4 + X) 3 9 ’ 
>> ( 66 ). 
If 
a 2 
X 
\ 2/3 
a+x 
\ 2 —X 2 187 + 110X-14X 2 4 
1 - : 9 - - 02 */, . xX2 9 
3 (4 + X) 
3 2 .7 (4 + X) 2 
28492 + 36723X+ 14160X 2 —686X 3 , 6 
3 4 .7 2 (4 + X) 3 9 ' 
It is easy to verify that the product of the three series is unity, as should be 
the case. 
The next step is to substitute for cf, f, f ... their values in terms of a 3 /?- 3 and its 
powers. In this way I find 
a 
X \ 2 / 3 
1 + X 
or 
7> 
a" 
X \ 2 ' 3 
1+X/ 
If 
a 2 
' X \ 2 ' 3 
1 + X 
‘ 5 (7 + X) a 3 25 (419 + 187X+llX 2 )a 6 
. 6(1+X)r 3 2 2 .3 2 .7 (1 +X) 2 r 6 
125 (99848 +74769X+ 16503X 2 +488X 3 ) a 9 
2 3 .3 4 .7 2 (1 +X) 3 Y- 9 
' 5(5 + 2X)a 3 25 (11 + 37X —X 2 ) a 6 
6 (1 + X) r 3 + 2 2 .3 2 .7 (1 +X) 2 r 6 
125 (23992 + 17895X+1587X 2 +178X 3 ) a 9 
2 3 .3 4 .7 2 (1 +X) 3 
j _ 5 (2 —X) a 3 25 (157 + 119X—11X 2 ) a 6 
6 (1+ X) r 3 2 2 .3 2 .7 (1 +X) 2 r 6 
125 (19114 + 23673X+ 11577X 2 —488X 3 ) a 9 
2 3 .3 4 .7 2 (1 + X) 3 
9 
9 • • * 
(67). 
By writing 1/X for X we obtain the formulae for the axes of the other ellipsoid, 
the numerical coefficients increase rather rapidly so that the series are useless 
unless a/r is small, and accordingly this method fails to give any result for Roche’s 
ellipsoid in limiting stability ; it is, however, useful for the problem of figures of 
equilibrium, as already stated. 
If we had relied on spherical harmonic analysis, we should only have obtained the 
terms in a 3 /r 3 . 
