160] Roche's Model 159 



Let s denote f>p/p, the excess condensation of the varied motion, so that 

 the bracket on the left-hand side of the above equation is equal to s. We 

 have 



dd> * 1 dp 2 dp 



a P = - ^ &P = f s - 

 op pop op 



Using 7 temporarily to denote the gravitation constant, we have 

 dX dY 8Z 



^ -- H r~ + = = 47T7P, 

 dx dy 9^ 



so that 



dx dy 

 Equation (415) accordingly becomes 



asr 



h -5 -- h -5 = 47T7/OS. 



(416), 



and is now seen to be a differential equation determining the condensation s 

 in the varied motion. 



Putting 7 = in this equation, and thereby neglecting the effects ot 

 gravitation, we are left with the well-known equation 



which simply expresses that any excess condensation s is propagated as a 

 wave with a velocity \f(dp/dp) relative to the moving jet. In this case any 

 displacement from the original motion can only give rise to small oscillations 

 about this motion, so that the motion is thoroughly stable. 



Restoring 7 and assuming for simplicity that p and dp/dp are uniform 

 throughout the jet, we find that a solution of the full equation (416) can be 

 obtained by taking s proportional to an exponential factor 



this solution representing waves of wave-length X projected with a velocity 

 q\/%7r. Certain boundary conditions must be satisfied in addition to the 

 differential equation (416). These may be taken to be that s shall vanish at 

 the two ends of the jet, say at x = and x = L 



On substituting the exponential factor into the differential equation (416), 

 we find that 



while the boundary conditions are satisfied if 



