THEOREMS ON BIFURCATION AND STABILITY, 



143 



on the numerical value of the sum of an infinite series; and from a compu- 

 tation of its first terms Darwin regards it as certain, though not algebrai- 

 cally proved, that the pear-shaped figure is stable. However, Liapounofif 

 has stated^ that these figures are unstable. If so, the line of completely- 

 stable figures terminates at this point, and as soon as a body has passed be- 

 yond it a slight disturbance will cause it to undergo radical changes of form 

 and perhaps break into many fragments. Even if the pear-shaped figures 

 are at first stable, they may become unstable as well as all figures which 

 branch from them long before fission occurs. Indeed, this now seems prob- 

 able, for Darwin has found,^ in a memoir on the figure and stability of a 

 liquid satellite, that a satellite loses its stability before it can be brought 

 near enough to its primary to coalesce with it. 



Consequently if a slightly viscous fluid mass were originally turning 

 slowly and had the form of a stable Maclaurin spheroid of equilibrium, and 

 if in some way greater and greater rates of rotation were gradually impressed 

 upon it without violently disturbing its figure, then we should see the series 

 of changes in its shape described by a point moving along the curve Ob to 

 the point b, then branching on to the line be of stable forms, again branching 

 at/, if the pear-shaped figures are stable, and continuing along lines of stable 

 figures until they terminate or until fission takes place. At any rate there 

 is no possible chance of fission until the change of shape has passed beyond 

 b, for up to this point there is secular stability and no branching. In fact, 

 we may feel assured that it can not occur until the shape of the mass has 

 passed at least to /. In order that we may see to what a remarkable extent 

 a rotating homogeneous fluid must depart from sphericity before there is a 

 possibility of fission starting, we give in figure 19 the most oblate section 



Fig. 19. 



of the ellipsoid belonging to the point/. Darwin has shown^ that its eccen- 

 tricity is 6 = 0.9386. The eccentricity of the other principal section through 

 the axis of rotation is e' = 0.6021, and the eccentricity of the principal section 

 perpendicular to the axis of rotation is e" = 0.9018. 



* Acad. Imp. des Sci. de St. P^tersbourg, 17, No. 3 (1905). 

 2 Phil. Trans., A, 206 (1906), pp. 161-248. 



^ On the pear-shaped figure of equiUbrium of a rotating mass of liquid. < Phil. Trans., A, 

 198 (1902), pp. 301-331. 



