36-39] 



The Rotational Problem 



39 



Jacob fs Ellipsoids 



39. Let us now examine the second alternative, represented by equation 

 (70) in 37. For these configurations a is no longer equal to b, so that the 

 integrals do not admit of integration in finite terms. They have been dis- 

 cussed by C. O. Meyer*, and also reduced to elliptic integrals and treated 

 numerically by Darwinf. 



It is found that the 'ellipsoids form one single continuous series ; they 

 are generally known as Jacobian ellipsoids, their existence having been first 

 demonstrated by Jacobi in 1834J. The maximum value of a> 2 /'27rp is found 

 to occur for the particular ellipsoid for which ab\ this value is 18712, 

 and the ellipsoid for which it occurs is one in which a = 6 = l*7!61c. This 

 configuration is also of course a Maclaurin spheroid, and so forms a point 

 of bifurcation on this latter series. It is the configuration printed in heavy 

 type in the table above. 



As we pass along the Jacobian series, the ratio a/b may be supposed to 

 vary continuously from to oo , and the point of bifurcation occurs when 

 a = b. The two halves of the series are however exactly similar, either one 

 changing into the other on interchanging a and b, so that we may legiti- 

 mately confine our attention to one half, say that for which a > b. We now 



* Crelle's Journ. 34 (1842). 



+ Proc. Roy. Soc. 41 (1887), p. 319 or Coll. Works, in. p. 118. 



$ Pogg. Ann. 33 (1834), p. 229. 



