141-145] General Theory 143 



The configurations near to the spherical one are spheroids of small ellip- 

 ticity, but the series will not remain spheroidal throughout its length. But 

 the far end of this series is again spheroidal, being in fact identical with the 

 Maclaurin spheroid for a mass of uniform density <r. Just as in the incom- 

 pressible problem, this is unstable for all displacements specified by sectorial 

 harmonic deformations of its boundary*. It follows that, on the series we 

 are considering, there must be points of bifurcation corresponding to all 

 sectorial harmonics. The general physical principles explained in 82 lead 

 us to expect with confidence that the first of these to occur will be that 

 corresponding to the second harmonic. At this point the circular cross- 

 section of the figure gives place at first to an elliptic cross-section of small 

 ellipticity, and the configurations on the new series are analogous to the 

 Jacobian ellipsoids. 



Further, the far end of the series analogous to the Jacobian series is 

 again identical with that in the incompressible problem, both as regards 

 configuration and stability, so that again this series must have the same 

 points of bifurcation as the Jacobian series. 



143. Almost identical remarks apply to the tidal problem. Again there 

 is a principal series of figures of revolution analogous to the tidal spheroids 

 examined in 49, and again these figures are strictly spheroidal at the two 

 extreme ends of the series. The stability of the end configuration of this 

 series a long drawn out line of matter is plainly the same as for the 

 incompressible mass, so that the same points of bifurcation must occur on 

 the series. 



144. All these statements obviously require slight modification in the 

 extreme case of cr 0, but except for this case it is clear that the general 

 arrangement of series and points of bifurcation will be very similar to that 

 in the incompressible problem. It ought again to be possible to construct a 

 diagram similar to that of fig. 7 (p. 50) ; the general arrangement will be 

 the same but the numerical values different, and the shape of the figures 

 will of course be different except at the extreme ends of the various series. 



Figures of equilibrium which take the place of the spheroidal figures of 

 the incompressible problem, whether rotational or tidal, may conveniently be 

 referred to as " pseudo-spheroids." Similarly figures which take the place of 

 ellipsoidal figures of equilibrium may be referred to as " pseudo-ellipsoids '' ; 

 these of course do not enter in the tidal problem, but occur in the rotational 

 and double-star problems. 



145. This general discussion does not touch the question of the stability 

 of the various branch series; this can only be determined by detailed 



* Of. Poincar^, Ada Math. 7 (1885), p. 259. 



