FIGURE AND STABILITY OF A LIQUID SATELLITE. 
201 
This is the equation to be satisfied by the axes of the ellipsoid. If we treat e and rj 
as zero, it is the same as that found by Roche. # 
* The form of this equation is so unlike Roche’s, that it may be worth while to prove the identity of 
the two. 
Roche writes his equations in the form 
st (t - s) 
u du 
«(1 ~ s ) 
i: 
u du 
*( 1-0 
j: 
u du 
(3 + A) t — As J o (1 + sit) (1 4- tu) R s+3 + Ajo(l+ sit) (1 4-if) 11 t + A J o (1 + it) (1 + tu) R 
where R 2 = (1 +it) (1 +su) (1 +tu), and s is the square of the ratio of the least to the greatest axis, and t 
the square of the ratio of the least to the mean axis. 
In my notation 
a 2 9 , ft 2 cos 2 y 
s = — = cos 2 y, t = -j- = — 5 -i . 
c 2 b 2 cos 2 ft 
* 9 
If we write us + 1 = 11 and change the independent variable from u to xp, we find 
sin 2 xp 
-r, , c , 2 cos 3 ft fv sin 2 xp (sin 2 y - sin 2 xp) 7 . 
• Roche s first integral = -t-t- — — -- ., -— dxp 
° sm 5 y cos 3 y Jo A 3 
second 
third 
2 cos ft fy sin 2 xp (sin 2 y - sin 2 xp) 
dip 
sin 5 y cos y J o cos 2 xp A 
_ 2 cos 3 fi fr sin 2 1 p (sin 2 y - sin 2 1 p) ^ , 
sin 5 y cos y I o cos 2 xp A 3 
I 
where A 2 = 1 - k 2 sin 2 xp. 
The coefficients are 
(A), 
st ( t-s) _ 
(3 4- A) t - As cos 2 ft [(3 + A) - A cos 2 ft] 
cos 4 y sin 2 ft 
«(1 - *) = 
s + 3 + A 
*( 1-0 
sir y cos 2 y 
3 + A + cos 2 y 
k' 2 sin 2 y cos 2 y 
(B). 
t + A cos 2 /J (cos 2 y + A cos 2 ft) 
Then Roche’s equations are equivalent to 
1st of (A) x 1st of (B) = 2nd of (A) x 2nd of (B) = 3rd of (A) x 3rd of (B). 
But the two equations are not independent, and I will only pursue the consideration of the form 
involving the 2nd and 3rd of (A) and (B). 
Now 
sin 2 xp (sin 2 y - sin 2 xp) _ sin 2 xp # tan 2 xp 
cos 2 xp A A ‘ 7 A 
sin 2 xp (sin 2 y — sin 2 xp) _ cos 2 ft sin 2 xp cos 2 y tan 2 xp 
cos 2 xp A 3 k' 2 A 3 k' 2 A 
and I have proved in (25) of the “ Pear-shaped figure, &c.” that 
fy sin 2 xp 
Si = 
= -A- r 
sin 2 y J t 
A 
dxp, 
A , 1 = Pi'Q , 1 = iisil f v Issli if, 
^ sin 2 y Jo A 
9l . = SlW _ i£2!V J p2 pdf. 
sm 2 y J o A d 
2 D 
VOL. CCVI.-A. 
