i60 Compressible and Non-Homogeneous Masses [CH. vn 



where n is any integer. Thus the different values of q are given by 



The vibration of lowest frequency is given by n = 1, and cf becomes negative 

 when I first reaches the value 



l=\\J - f (419). 



V 7/3 dp 



This fixes the first dynamical point of bifurcation ; as I increases more and 

 more points of bifurcation occur, the complete set being given by 



Thus as I increases, one vibration after another loses its stability. The 

 initial unstable motion of any vibration is one in which the matter of the jet 

 tends to collect into nuclei or bunches at the nodes of the wave. Or, alter- 

 natively, we may consider that a series of furrows tends to form in the jet, 

 and that these get continually deeper. After passing the first point of bifur- 

 cation one furrow tends to form, namely a furrow between the jet and the 

 main body ; after passing the next point of bifurcation two furrows begin to 

 form, and so on. 



161. Clearly the formation of 1, 2, 3 ... furrows in succession in this 

 problem is very closely analogous to the formation of 1, 2, 3 ... furrows in 

 succession which occurs when the incompressible mass passes points of bifur- 

 cation corresponding to harmonics of orders 3, 4, 5 .... 



Although the formation of furrows in these two problems is closely analo- 

 gous, it would be a mistake to suppose that the two solutions we have obtained 

 merge gradually into one another as the compressibility of the primary mass 

 gradually changes. We may notice that the breaking up of an incompressible 

 mass takes place independently of its size, whereas the breaking up of the 

 jet of matter formed from the atmosphere in Roche's model will only take 

 place when the system is beyond a certain size. 



162. Further insight into the motion will be obtained from a consideration 

 of the composite model already discussed in 154. We suppose the primary 

 to consist of an incompressible nucleus of volume v^and density p , surrounded 

 by an atmosphere of volume V A and negligible density. Under tidal forces the 

 configuration of the nucleus will be exactly the same as if the atmosphere 

 were non-existent, while the atmosphere will be bounded by one of the equi- 

 potentials surrounding this nucleus. 



It will be sufficient to consider the simplest case in which M'jM is infinite. 

 In this case the total gravitational potential is 



