190, 191] Spherical Mass of Gas 193 



191. The ratio of the specific heats 7 being assumed uniform throughout 

 the mass, let us introduce a new quantity k denned by 



p = kpv ................................. (513). 



Combining this relation with the general equation (507) w^ obtain 



. ...(514). 



m 



If a configuration is one of adiabatic equilibrium, k will of course have a 

 uniform value throughout. We shall now see that only those configurations 

 are mechanically stable in which k either stays constant or increases at every 

 point as we pass from the centre to the surface. 



To see this, let us fix our attention on any two small elements of gas in 

 different parts of the mass. Let the first be of mass M lt volume S^ and 

 density p lt and let it be at a point P l at which the pressure is p l and the 

 value of k is fa. Then 



Let the same symbols with suffix 2 refer to the second element. 



Let us consider the process of interchanging these two elements, the 

 remainder of the gas remaining undisturbed. To do this, the element M l 

 must be compressed or expanded to a volume Bv 2 , so that its new density 

 will be MJ&Vt. Let us suppose this contraction or expansion to take place 

 adiabatically, then the final pressure will be 



<*()' 



If this element can be placed in the cavity Bv 2 at P 2 without creating a 

 disturbance in the remainder of the gas, the pressure just calculated must be 

 equal to the equilibrium pressure at P 2 , and this is given by 



Jf. 



P = 



The pressures are accordingly equal if 



MV = MV (516). 



This is the condition that M l can be fitted into the place of M z without 

 disturbance ; since it is symmetrical, it is also the condition that M 2 can be 

 fitted into the place of M l without disturbance. Thus if M l and M 2 are chosen 

 so that this condition is satisfied, the two elements can be interchanged with- 

 out any work being done except that done against the gravitational field. Of 

 the two masses, let M l be the one originally nearer to the centre. A con- 

 dition for the stability of the original configuration is clearly that the work 

 done in any interchange such as that just considered shall be positive, and 

 j. c. 13 



