470 x STATICS OF DEFORMABLE BODIES. 



The course of the function 



is shown in Fig. 150, from which 

 it is evident that if 



< 0.22467 



Fig. 150. 



there are two values of I satisfying 

 equation 38) and accordingly two 

 possible ellipsoids of rotation. If 

 on the contrary 



> 0.22467 



" 



no possible ellipsoid of rotation is a figure of equilibrium. 



When co is very small one of the values of 'L tends to zero and 

 the other to infinity, that is, one of the ellipsoids is a sphere, the 

 other a thin disc of infinite radius. 



In the case of the earth using the value of yQ of 123 and 



of co of 149. - - = .00230. and the smaller of the two values 



7 %icyg 



of A coincides most nearly with the actual facts 1 ), giving 

 A 2 =0.008688, e = ^- 



The actual ellipticity being however ^^ we can only conclude that 



^jy y 



the earth when in its fluid state was not homogeneous. 

 The transcendental equation 36) written out is 



40) 



HD 



-e 2 ) C\-*- 

 >J la'+ 





or otherwise 



41) (V - 



du 



{<**+ 



=0. 



Besides the solution 6 = c there is another given by putting the 

 integral equal to zero. When a = 0, the integrand and consequently 



the definite integral is negative, when a = - - the integral is 



j/& 2 -t-e 2 



positive. There is accordingly a real value of a which satisfies the 

 equation and there is an ellipsoid with three unequal axes which is 

 a possible figure of equilibrium, if co lies below a certain limit. 

 This result was given by Jacobi in 1834. For further information 

 on this subject the reader is referred to Thomson and Tait, Natural 

 Philosophy, 771778. 



1) Tisserand, Traite de Mecanique Celeste, Tom. II, p. 91. 



