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.