203, 204] General Theory 205 



In an actual mass of gas in which G denotes the ratio of the specific heats 

 at any point, p and p may be supposed connected by the relation 



in which k will vary from point to point. We accordingly firict " 



9 ]?! = Q + 81 gfc 

 9 log p d log /? ' 



If we use the symbol 7 to denote 3 logp/3 log/o, this becomes 



(533). 



We have seen that, as we pass along the radius of an actual mass of gas 

 from the centre to the edge, k will continually increase, or at most remain 

 constant, while p will continually decrease. Thus k and p change in opposite 

 senses, so that d log k/d log p will be negative at every point of the gas, and 

 the effect of the non-constancy of k will be to decrease the value of 7. 



The same result may be obtained by noticing that the eqiiilibrium of an 

 isothermal mass of gas is the same as that of a gas in adiabatic equilibrium 

 with 7 = 1, while in an actual mass of gas conditions are such that the equi- 

 librium is intermediate between adiabatic and isothermal equilibrium. 



In our study of the " adiabatic " model, we found that the series of 

 equilibrium configurations was of the same general type for all values of 7 

 less than 2-J. In an actual mass of gas, 6r, the true ratio of the specific heats 

 must be less than If, and the value of 7, as determined by equation (533) 

 must be still less on account of the non-constancy of k. Thus it seems 

 permissible to assume that the sequence of configurations in an actual gas 

 would be of the same type as those in an "adiabatic " mass in which y< 2. 

 And this series of configurations, as we saw, consisted of spheroids when the 

 rotation was small, these giving place to pseudo-spheroids for larger rota- 

 tions, and ultimately giving place to a lenticular figure with a sharp edge 

 from which matter was thrown off. 



The effect of the sinking of the heavier chemical elements to the centre 

 is easily allowed for. It results in an excess of central condensation of mass, 

 and this may be allowed for by supposing the mass to approach more nearly 

 to Roche's model than would be the case if the gas were of uniform com- 

 position. We have just seen that the motion of the gas, even without 

 allowing for this excess of central condensation, will in its main features be 

 the same as that of Roche's model. The resemblance of the motion to that 

 of Roche's model will be still closer when central condensation of mass is 

 taken into account. 



