192-194]
Ellipsoidal Configurations
211
Jacobi's Ellipsoids.
193. Let us now examine the configurations which constitute the second
linear series, determined by equation (191*2). In these configurations a is
no longer equal to b, and the integrals do not admit of integration in finite
terms. They have been discussed by C. O. Meyer*, and also reduced to
elliptic integrals and treated numerically by Darwin t*
The ellipsoids are found to form one single continuous series. The
maximum value of &> 2 /27r7p is found to be 0*18712; this occurs for the parti
cular ellipsoid which a = b = 1*7161 c. This configuration is also of course a
Maclaurin spheroid, and so forms a point of bifurcation on this latter series.
It is the configuration printed in heavy type in the table above.
As we pass along the Jacobian series, the ratio a/b may be supposed to
vary continuously from 0 to oo, and the point of bifurcation occurs when
a — b. The two halves of the series are, however, exactly similar, either one
changing into the other on interchanging a and b, so that we may legitimately
confine our attention to one half, let us say that for which a >b. We now
regard the series of Jacobian ellipsoids as starting at the value a = b (the
point of bifurcation), and the ratio a/b continually increases from 1 to oo as
we pass along the series. The following numerical values are given by
Darwin $:
Table XVII. Jacobian Ellipsoids.
a
r 0
b
r 0
c
ro
w 2
2iryp
Angular
Momentum
1-1972
1-1972
•6977
•18712
•30375
1-216
1-179
•697
•1870
•304
1-279
1123
•696
•186
•306
1-3831
1-0454
•6916
•1812
•3134
1-6007
•9235
•6765
1659
•3407
1-88583
•81498
65066
14200
•3898
1-899
•8111
•6494
•1409
•3920
2-346
■7019
•6072
•1072
•4809
3-1294
•5881
•5434
•0661
•6387
5-0406
•4516
•4393
•0259
1-0087
00
0
0
0
oo
194. We have now mapped out the various configurations on the two
linear series of ellipsoidal configurations—the Maclaurin spheroids and the
Jacobian ellipsoids—and the stability of these configurations can be investi
gated by the methods already explained.
* Crelle's Journ. xxxiv. (1842).
+ Proc. Roy. Soc. xli. (1887), p. 319, or Coll. Works, hi. p. 118.
$ Coll. Works, hi. p. 130.