157-160] Roche's Model 157 



be neglected and equation (409) requires to be modified accordingly. We 

 can, however, see that the correction to this equation cannot be large, being 

 of the order of 5 per cent, at most*, so that equation (409) may be regarded 

 as giving a tolerably good approximation for all ratios of M ' to M. 



Thus both when M r 2M and when M'jM= oo , it appeal's I/hat hetero- 

 geneity, even of the very extreme kind now under consideration, has only 

 a very slight influence upon the critical value of R. 



159. We have accordingly found that the series of equilibrium configu- 

 rations stops for about the same values of R as those for which the corre- 

 sponding series became unstable in the incompressible problem. In the 

 incompressible problem it was an easy matter to determine the dynamical 

 motion which occurred when statical motion was no longer possible. We 

 found that at first the primary rapidly elongated itself, while still retaining its 

 spheroidal form. After a time this motion was disturbed by the occurrence of 

 what we have called dynamical points of bifurcation ; furrows formed round 

 the figure and these seemed likely to result in its ultimate fission into a 

 number of detached masses. 



In the compressible problem now under consideration the dynamical 

 motion is, as we shall see, very similar to that just described. Consider first 

 the simplest case in which M'/M = 00. In this case, as soon as the critical 

 point is reached, the equipotential by which the mass is bounded opens sym- 

 metrically at both ends and matter is ejected. This matter will form two 

 long symmetrical jets or arms and the elongation of these arms corresponds 

 fairly closely to the elongation of the spheroid in the incompressible figure 

 of equilibrium. We shall now see that, during this process of elongation, 

 dynamical points of bifurcation will oecur, very much in the same way as in 

 the incompressible problem. 



160. The motion of the ejected streams of matter will of course be deter- 

 mined by the usual hydrodynamical equations which may be expressed in 

 the form 



3 2 # v I dp 



-~- 7 = X - - ^- etc., 

 ot 2 p ox 



in which all the symbols have their usual meanings. As in 140, let us put 



f = *o, 



thereby assuming that p is a function of p only, and let us denote the com- 

 ponents of acceleration of the particle which is at x, y, z at time t 

 Then the equations of motion become 



tc (411). 



* Cf. Mem. R.A.S. 72, pp. 1014. 



