142 THE TIDAL PROBLEM. 



III. POINCARE'S THEOREMS RESPECTING FORMS OF 

 BIFURCATION AND EXCHANGE OF STABILITIES. 



In a memoir^ remarkable for its powerful methods and important results, 

 Poincare has proved the existence of an infinite number of other forms of 

 equilibrium. He considered the equations for equihbrium as functions of the 

 parameter w. For a definite value of to, as w = a>o, they have a certain 



number of solutions. For example, for -^^<0.22AQ1 there is a solution 



on Oa and one on Pa of figure 17. If -^^ < 0.18709 there are also solutions 



on he and hd. If for io = io^ two or more solutions unite and do not vanish 

 as ix) passes through lo^, then the figure of equilibrium corresponding to w^ 

 is a/onn of bifurcation. If after uniting they vanish, the figure is a limit 

 form. Thus, in figure 17, h belongs to a form of bifurcation, for at this point 

 the Maclaurin spheroids and Jacobian ellipsoids are identical. The point a 

 belongs to limit form, fcr at this point the two series of Maclaurin spheroids, 

 Oa and Pa, unite and vanish. Likewise the point h belongs to a hmit form 

 for the series hd and he. 



Poincare showed in the work cited that there are no forms of bifurcation 

 corresponding to points on the curve Oh, but that there is an infinite num- 

 ber of them on haP. He proved also that there is an infinite number of 

 them on 6c and hd between the point h and the axis co = 0. That is, in addi- 

 tion to the spheroids and ellipsoids of equilibrium an infinite number of 

 other forms exist. The first one on he is at /, and its deviation from the 

 Jacobian eUipsoid to the first order of small quantities depends upon the 

 third zonal harmonic with respect to the greatest axis of the ellipsoid. It 

 is the pear-shaped figure referred to above. Since it is unsymmetrical with 

 respect to the axis of rotation, there are really two similar figures, differing 

 by 180° in orientation, just as the two series of Jacobian ellipsoids differ by 

 90° in orientation. There is of course a precisely similar series on hd. 



If two real series of figures of equilibrium, A and B, cross, and if before 

 crossing A is stable and B unstable, then after crossing A is unstable and B 

 has at least one degree less of instability. Poincare has also proved an en- 

 tirely similar theorem in periodic solutions of the problem of three bodies.=* 



All the spheroids corresponding to points on the curve Ob of figure 17 are 

 completely and secularly stable. At h the spheroids lose their stability but 

 the branching Jacobian ellipsoids are stable. They remain stable until /is 

 reached. It is an interesting question whether the pear-shaped figures are 

 stable or unstable. Poincare threw the determination of the answer to the 

 question into a form capable of numerical treatment,^ and Darwin has 

 made an elaborate and detailed discussion of i t." The rigorous answer turns 



» Sur r^quilibre d'une masse fluide anim^e d'un mouvement de rotation. < Acta Mathe- 



matica, 7, 1885, 259-380. , , „, . ^.-1.0 qa? qaq 



2 Les M6thodes Nouvelles de la Mecamque Celeste, 3, pp. 347-349. 



3 Sur la stabilite de I'equilibre des figures pyriformes affect^es par une masse flmde en 

 rotation.<Phil. Trans., A, 198 (1902), pp. 333-373. , v -^ 



*The stability of the pear-shaped figure of equilibrium of a rotating mass of liquid. 

 <Phil. Trans., A, 200 (1903), pp. 251-314. 



