FIGURE AND STABILITY OF A LIQUID SATELLITE. 229 
Interpolating from these we find four values satisfying/(sin -1 y, k) = 0, namely:— 
/(77°, 57°'5846) = 0, f(7 8°, 59°-4798) = 0, /(79°, 6R-3744) = 0, /(80°, 63°-2806) = 0. 
With these solutions I find 
sin 1 k 
rf a 
77° 
2-467860 
•<r 
00 
o 
2-458191 
79° 
2-455446 
o 
O 
00 
2-460289 
By formulae of interpolation the minimum of r occurs when sin -1 k - 78°"8756. Then 
by a second interpolation this value of k corresponds with y = 61°'1383, and the 
minimum value of r is 2‘45539. We may take then y = 61° 8'"3, k = sin 78° 52'"5, 
whence ft = 59° 14 /- 5. Since cos y = 0 4827, cos /3 — 0’5114, the three axes of the 
ellipsoid are proportional to 10000, 5114, 4827; Roche gave the ratios 1000, 496, 
469, and the radius vector as 2‘44, in place of 2’45539. 
Turning now to the second solution, I solved the problem by means of the auxiliary 
tables in two ways, namely, for y = 60°, 61°, 62° and also for y = 57°, 59°, 61°, 63°. 
They led to virtually identical results, viz., that the minimum of r is 2'45521, 
corresponding to y = 61° 8'"4, sin -1 k — 78° 52 /, 0. 
Finally the solution for Roche’s limit and for the ratio of the axes of the ellipsoid 
in limiting stability may be taken to be as follows :— 
y sin -1 k cos y cos /3 r \a 
61° 81' 78° 52' 0-4827 0-5114 2*4553, 
with uncertainty of unity in the last place of decimals in r and of half a minute of 
arc in sin -1 k. 
We must next consider the modified form of Roche’s problem, in which the large 
body or planet yields to centrifugal force and becomes an oblate ellipsoid of revolution. 
The approximate formulae of § 19 show that when X = oo or when X = 0, 
Hence in this case 
1 = 1, m = yT, n 
2 2 5 
2 2 4" 
3 
4 
A 2 5. 
2 2 4 
The solution of the modified problem can only differ slightly from that just found 
when the planet is spherical, and therefore we may compute £ with sufficient accuracy 
by means of the values of a/r already found. I accordingly computed £ for 
y = 60°, 61°, 62°, and found that in each case £ was very nearly equal to 0‘0088. 
Now it is proved in the footnote to § 10 that, when X = 0 and when the planet 
yields to centrifugal force, e — t) = £; as the value of £ is found with good approxi¬ 
mation, it is easy to compute r for these three values of y. I thus find that in the 
