80 Pear-shaped Configurations of Equilibrium [OH. v 



The condition that equation (208) shall be satisfied for any value of P is 

 exactly the same as that Poincare's " coefficient of stability " (cf. 23) shall 

 vanish, and Poincare gives, in terms of harmonic analysis, a very 'full account 

 of the conditions under which this can occur*. The discussion is too long to 

 insert here, and too intricate to summarise ; the principal results obtained 

 are the following : 



(i) The only type of harmonic for which equation (208) can be satisfied 

 (or Poincare's " coefficient of stability " vanish) is the zonal type. 



(ii) As the ellipsoid lengthens, the first harmonic for which equation 

 (208) can be satisfied is the third zonal harmonic, and beyond this point the 

 equation can be satisfied in turn for zonal harmonics of all orders from 

 4 to x. 



Poincare's discussion, being concerned only with the rotational problem, 

 deals only with the Jacobian series of ellipsoids, but for the reasons stated 

 above, is equally applicable to all our ellipsoidal configurations. 



It follows that on each of the series represented in fig. 7 (p. 50), as we 

 proceed from S to (JT) X , we must pass an infinite number of points of 

 bifurcation. Each corresponds to some value of P in equation (201), and 

 the different values of P are all zonal harmonics ; the first point of bifurca- 

 tion is that for which P is of the third degree. 



82. The necessity of this result can easily be" seen from physical con- 

 siderations, although it would probably not be easy to construct a rigorous 

 proof. 



Let W denote the total potential energy of the fluid mass under its own 

 gravitational forces and the statical field of force (fictitious or otherwise) 

 arising from rotation and tidal action. When a displacement occurs such 

 that the equation of the boundary is altered by the addition of the term eP 0) 

 as in equation (201), let the new potential energy be W + &W. 



Since the original configuration was one of equilibrium, SW will neces- 

 sarily be of the second order of small quantities, arid the original equilibrium 

 will have been unstable if 8W can be made negative for any value of I\. 



If m is the mass of any particle of the fluid, and V its potential in the 

 original configuration, the value of W can be put in the form 



Now let a displacement occur such that the typical particle of mass m is 

 moved to a position at which the potential in the old configuration was V, 



* L.c., 10 and 12. See also Schwarzschild, Inaug. Dissert. Mi'uichen (1896), and Darwin, 

 Coll. Works, m, pp. 302, 307. 



