210 The Configurations of Rotating Liquid Masses [ch. viii 
Elimination of 6 gives 
r ao 
-■<*-*> . KÈ ll! ' 2 ' 3 '- 
2^7 P j o 
Since <o 2 must be positive, a 2 must be greater than c 2 , so that the spheroids 
are all oblate. When a — b, the integral on the right admits of evaluation in 
finite terms, the equation then assuming the form 
s^ e - 3 {?- 0 ( 192 ' 4) ' 
where e is the eccentricity of the spheroid, defined by 
e i = 
a 2 — c 2 
or 
We notice that the eccentricity depends only on the ratio a) 2 /2Tryp. 
Further, when <w 2 /2-7ry/? is given, the eccentricity is fixed, but not the size of 
the spheroid; a spheroid of any size is a possible figure of equilibrium if its 
eccentricity has the value given by equation (192*4). The following table of 
values of w^^iryp and e is compiled from values given by Lamb*, Darwinf 
and Thomson and Tait+. 
Table XVI. Maclaurins Spheroids. 
e 
a K 
c/r 0 
w 2 
2iryp 
Ang. Momentum/2!i^ 0 
0 
1-0000 
1-0000 
0 
0 
*1 
1-0016 
•9967 
•0027 
•0255 
•2 
1-0068 
•9865 
•0107 
•0514 
*3 
1-0159 
•9691 
•0243 
•0787 
*4 
1-0295 
•9435 
•0436 
•1085 
*5 
1-0491 
•9068 
•0690 
•1417 
*6 
1-0772 
•8618 
•1007 
•1804 
*7 
1-1188 
•7990 
•1387 
•2283 
*8 
1-1856 
•7114 
•1816 
•2934 
*81267 
11972 
•6977 
•18712 
•30375 
*9 
1-3189 
•5749 
•2203 
•4000 
*91 
1-341 
•5560 
•2225 
•4156 
*92 
1-367 
•5355 
•2241 
•4330 
•93 
1-396 
•5131 
■2247 
•4525 
*94 
1-431 
•4883 
•2239 
•4748 
*95 
1-474 
•4603 
•2213 
•5008 
•96 
1-529 
•4280 
•2160 
•5319 
•97 
1-602 
•3895 
•2063 
•5692 
•98 
1-713 
•3409 
•1890 
•6249 
■99 
1-921 
•2710 
•1551 
•7121 
1-00 
00 
0 
0 
CO 
The values of e range from 0 to 1*00, so that every oblate spheroid is a 
configuration of equilibrium for some value of &> 2 /27ry p. On the other hand, 
the values of (o 2 /27ryp never exceed 0*225, so that there are no spheroidal 
configurations of equilibrium when to 2 > 0*225 x 2 - 777 /?. 
* Hydrodynamics (4th Ed.), p. 673. 
t Proc. Roy. Soc. xli. (1887), p. 319, or Coll. Works, hi. p. 119. £ Nat. Phil. § 772.