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.