314-315] JACOBl'S ELLIPSOID. 585 



315. To ascertain whether an ellipsoid with three unequal 

 axes is a possible form of relative equilibrium, we return to the 

 conditions (5). These are equivalent to 



-& 2 ) = ............ (10), 





If we substitute from Art. 313, the condition (10) may be 

 written 



The first factor, equated to zero, gives Maclaurin's ellipsoids, 

 discussed in the preceding Art. The second factor gives 



-r (13 >> 



which may be regarded as an equation determining c in terms of 

 a, 6. When c 2 = 0, every element of the integral is positive, and 

 when c 2 = a 2 6 2 /(a 2 4- 6 2 ) every element is negative. Hence there is 

 some value of c, less than the smaller of the two semiaxes a, b, 

 for which the integral vanishes. 



The corresponding value of n is given by (11), which takes the 

 form 



, r \d\ 



=abc T^rrrwiJTvnr ( 14 )> 



so that n is real. It will be observed that as before the ratio 

 n^/^Trp depends only on the shape of the ellipsoid, and not on its 

 absolute size. 



The possibility of an ellipsoidal form with three unequal axes 

 was first asserted by Jacobi in 1834*. The equations (13) and 

 (14) were carefully discussed by C. O. Meyerf, who shewed that 

 when a, b are given there is only one value of c satisfying (13), 

 and that, further, n*/27rp has its greatest value (1871), when 

 a = 6 = l'7!61c. The Jacobian ellipsoid then coincides with one 

 of Maclaurin's forms. 



* "Ueber die Figur des Gleichgewichts," Pogg. Ann., t. xxxiii. (1834); see also 

 Liouville, " Sur la figure d'une masse fluide homogene, en e*quilibre, et dou^e d'un 

 mouvement de rotation," Journ. de VEcole Polytechn., t. xiv., p. 290 (1834). 



t " De aequilibrii formis ellipsoidicis," Crelle, t. xxiv. (1842). 



