234 
SIR G. H. DARWIN ON THE 
§ 22. Harmonic Deformations of the Ellipsoids. 
When X is infinitely small, so that the liquid satellite is infinitely small, the 
harmonic deformations are evanescent, and the same is true when X is infinitely great. 
We saw in § 14 that there was reason to suppose that Roche’s ellipsoid in limiting 
equilibrium might be more markedly deformed for values of X midway between zero 
and unity than for the latter value. I therefore determined the harmonic deforma¬ 
tions in the three cases X = 0"4, 07, l'O. 
The formulae for the ellipticities f, f 2 , f, f 5 are given in §§ 14, 16, 17 and their 
values may be found for the ellipsoids in limiting stability tabulated in the last 
section. It appears that in every case the amount of deformation is small, and 
therefore it was sufficient to compute the normal deformation at the two extremities 
of the c semi-axis, that is to say, at the points nearest to and most remote from the 
other ellipsoid. At these points the normal displacement outwards may be denoted 
Sc, with numbers affixed thereto so as to indicate to which harmonic it is due. 
The results may be given in a tabular form, but it may be well to remark that f£ 
(second tesseral of the third harmonics) was only computed in two instances, because 
its effect was obviously quite negligible. For the same reason f 2 , f\ f 2 , f 5 l were 
not computed at all. The following are the computed values of the ellipticities :— 
X. 
0-4. 
0-7. 
1-0. 
A 
0-161 
0-234 
0-163 
A 2 
-0-127 
-0-102 
? 
A 
0-0042 
0-0066 
0-0059 
A 
0-0009 
0-0024 
0-0033 
We saw reason to suppose that the higher harmonics would be relatively more 
important for the larger values of X, and there is evidence of the general correctness 
of this view. 
The values of the ellipticities afford 
us no idea 
of the amount of the normal 
correction, and I therefore 
proceed to tabulate the values of Sc, the prolongations of 
the c semi-axis. 
Towards larger ellipsoid 
X. 
0-4. 
0-7. 
1-0. 
Sc 3 
0-0193 
0-0283 
0-0194 
§c 3 2 
0-00039 
0-00068 
2 
Sc 4 
0-00472 
0-00886 
0-00911 
Sc b 
o-ooiio 
0-00384 
0-00629 
Total Sc 
0-0255 
0-0417 
0-0348 
c 
0-8433 
0-9006 
0-9270 
c + Sc 
0-8688 
0-9423 
0-9618 
Sc 
c 
1/33 
1/22 
1/27 
