ELLIPSOIDAL FIGURES OF EQUILIBRIUM. 139 



II. THE ELLIPSOIDAL FIGURES OF EQUILIBRIUM OF 

 ROTATING HOMOGENEOUS FLUIDS. 



For the applications which follow it will be necessary to review briefly 

 the facts regarding the spheroidal and ellipsoidal figures of equilibrium and 

 their conditions of stability. 2 



Maclaurin* has shown that for very small values of „ ,., there are 



two ellipsoids of revolution which are figures of equilibrium, one of them 

 being nearly spherical and the other very oblate, the limits for a» = being 

 respectively the sphere and infinite plane. For greater values of this quan- 

 tity, the figure corresponding to the former is more oblate and that corre- 



sponding to the latter is less oblate. For ^ 72 =0.22467 . . . the two 



2 Ztzk a 



figures are identical. For ^ p > 0.22467 . . . there is no ellipsoid of 

 revolution which is a figure of equilibrium. 



2 



Jacobi has shown^ that if ^ ^ < 0.18709 . . . there is an ellipsoid of 



three unequal axes satisfying the conditions for equilibrium. When this 

 quantity is very small, the axis of rotation and one other are very short and 

 nearly equal to each other, while the third is relatively very long. With 

 greater values of this quantity the shorter axes are longer and the longest 



axis IS shorter. For ^ ^ =0.18709 . . . the figure becomes an ellipsoid 

 of revolution and is identical with the more nearly spherical Maclaurin 



2 



spheroid. For ^ ,, > 0.18709 . . . the Jacobian ellipsoids of revolution 



do not exist. 2 



(jj 



In the case of the Maclaurin spheroids the relation between ^ ,3 and 

 the eccentricity, e, of an axial section is given by the well-known equation^ 



ay" 



27:k 

 where 



e 



-=-^|-^^^tan-i.l-3| =<P(X) (1) 



; = 



■e^ 



^Jl-i 



It follows from these equations that for /I = we have 



^W^O M0=0 e = iA = l ^ = 1^ = 



dA de de d/. de 



and that for ^= cc we have 



^(^)=0 e = l ^^"^ 



' Treatise on Fluxions, Edinburgh, 1742. 

 ' Letter to the French Academy, 1834. 

 ^Tisserand, Mecanique Celeste, 2, Chap. VI. 



