232 
SIR G. H. DARWIN ON THE 
K = sin 51° 33'-5. The last step is to compute the axes of the two ellipsoids from 
the values of k, y, K, I\ 
Of course the numbers set out above make no claim to absolute accuracy, but the 
results tabulated below are, I believe, substantially correct. 
The unit of length employed is the radius of a sphere whose mass is equal to the 
mass of the whole system. If it were preferred to express the results in terms of 
the mean radius of the larger body, all linear results would have to be multiplied 
by (1 + X) 1/3 . 
We may now collect the results in a tabular form, as follows :— 
Solutions for Roche’s Ellipsoid in Limiting Stability. 
The 
unit of length is the radius of a 
sphere whose mass is equal to the sum of the 
masses, be., ctbc + ABC = 1 , and 
abc 
ABC 
= X. 
7- 
sin -1 k. 
a. 
l. 
c. 
r. 
sin -1 K. 
A. 
B. 
C. 
T. 
0 
61° 12' 
78° 50' 
0 -482-f- oo 
0 ’511 -h CD 
1-0 -f-oo 
— 
— 
0-942 
1 -030 
1 -030 
2-457 
0-4 
48° 13' 
69° 39' 
0-562 
0 -603 
0-843 
34° 25' 
51° 34' 
0-815 
0-886 
0-988 
2 -485* 
0-5 
46° 40' 
68° 12' 
0 -597 
0-642 
0-870 
35° 59' 
54° 30' 
0-792 
0-860 
0-979 
2 -484 
0-6 
45° 5' 
66° 43' 
0 -627 
0-G74 
0-888 
37° 14' 
56° 41' 
0-772 
0-836 
0-969 
2-490 
0-7 
43° 38' 
65° 20' 
0-652 
0-701 
0-901 
38° 9' 
58° 18' 
0-753 
0-815 
0-958 
2 -497 
0-8 
42° 26' 
64° 4' 
0-673 
0-725 
0-912 
38° 57' 
59° 39' 
0-737 
0 -796 
0-947 
2 -502 
0-9 
41° 25' 
62° 58' 
0-691 
0 -744 
0-921 
39° 40' 
60° 47' 
0-722 
0-778 
0 -937 
2 -508 
1 -o 
40° 15' 
61° 43' 
0-708 
0-762 
0 -927 
40° 15' 
61° 43' 
0-708 
0-762 
0-927 
2 -514 
The cases X = 0'4, 0"7, 1*0 are illustrated by figs. 2, 3, 4. The meaning of the 
dotted lines near the vertices of the smaller ellipsoid will be explained in the next 
section. 
The distance r—(c+C) is the interval between the vertices of the two ellipsoids; 
the following are the values, using, however, more places of decimals than are 
tabulated above :— 
A. 
r-(c + C). 
A. 
r- (c + C). 
0 
1-030 
0-7 
0-638 
0-4 
0-653 
0-8 
0-643 
0-5 
0-635 
0-9 
0-650 
0-6 
0-633 
1-0 
0-660 
It is remarkable how very nearly constant the intervening space remains throughout 
a large range in the values of X. 
* The values r = 2-485 for A = 0 - 4 and r = 2’484 for A = 0-5 represent 2• 4848 and 2-4S44 
respectively; it is probable that the last significant figure in the former is a little too large and in the 
latter too small, and that it might have been more correct to invert the 2 • 485 and 2 • 484 in the table. 
I give the result, however, of the computation. 
