230 
SIR G. H. DARWIN ON THE 
modified problem, minimum radius vector, and therefore limiting stability, occurs 
when r = 2’457, y = 61° 12', * = sin 78° 50', /3 = 59° 17'; the axes of the large body 
are determined by the approximate formulae of § 10 to be — = — = 1-0304, — = 0-9418. 
a a ’a 
It appears then that the yielding of the planet to centrifugal force makes very little 
difference, as was to be expected. 
These results are included in the table given below of results for solutions of the 
modified problem of Roche with finite values of X. 
§ 21. Roche’s Ellipsoidal Satellite, of finite mass, in limiting stability, the planet 
being also ellipsoidal. 
this is the problem which I describe as the “modified” problem of Roche. It 
seemed unnecessary to carry out the computations for the smaller values of X, since 
they are sufficiently represented by the case of the infinitesimal satellite where X is 
zero. I therefore begin with the case of X = 0-4 and pass on to X = On, 0‘6, 0 7, 
0-8, 0-9, 1-0. 
It seems well to describe the process folloAved in one case as a type of all. It was, 
in general, possible either by extrapolation from neighbouring values of X, or by mere 
guessing, to begin with some values of £, e, rj and their co-relative functions E, H for 
the larger body, which were somewhere near the truth. With these we could compute 
r > K ’ r ’ K with fair approximation; thence values of £, e, y, E, H could be calculated 
with close accuracy and the computation could be repeated. It was, of course, a 
mattei of conjecture as to what initial values of y would be found to embrace the 
region of minimum angular momentum. 
I will non describe the process for X = 0"4. Passing over the preliminary stages 
m which fairly good values were found, we begin with the following conjectural 
values:— 
y- 
46\ 
48°. 
50°. 
r. 
32°. 
34°. 
36°. 
sin -1 k 
68° 24'-8 
69° 32'-0 
70° 41'-3 
sin -1 K 
50° 12'-0 
51° 18'-8 
52° 34'-9 
r 
33° 13'"9 
34° 18'-8 
35° 18'-5 
7 
43° 50'-9 
47° 24'-4 
51° 34'T 
sin -1 K 
50° 52'-0 
51° 30'T 
52° 7'*5 
sin -1 K 
67° 15'-0 
69° ll'-8 
71° 37'-0 
log r 
0-40245 
0-39594 
0-39060 
log r 
0-41071 
0-39775 
0-38726, 
whence I compute 
i 
0-056259 
0-064863 
0-074219 
t 
0-047883 
0-062231 
0-082010 
6 
0-045435 
0-048102 
0-050256 
E 
0-140179 
0-174168 
0-218540 
V 
0-32004 
0-36426 
0-40838 
H 
0-40539 
0*52564 
0-69272. 
By means of these and the auxiliary tables I find 
sin 1 k 
68° 24'-6 
69° 31'-8 
70° 40'-5 
sin -1 K 
50° ll'-8 
51° 18'-8 
52° 33'-8 
log r 
0-40240 
0-39591 
0-39046 
log r 
0-41068 
0-39768 
0-38693. 
