194 The Evolution of Gaseous Masses [CH. vm 



this requires that M l shall be greater than M z , or again, from equation (516), 

 that h\ shall be less than k 2 . 



Thus the -necessary and sufficient condition for stability as regards inter- 

 change of places of the different elements or layers of the gas is that k shall 

 increase continuously from the centre to the surface. 



If, in any configuration, dk/dr is negative over any range, convective cur- 

 rents will be set up and the various layers will change places, until a new 

 configuration is formed in which dk/dr will be positive or zero everywhere. 

 And if steady agencies are at work tending to depress the value of dk/dr over 

 any range to a value below zero, a steady system of convection currents will 

 be set up of amount just sufficient to prevent dk/dr from falling below zero. 

 The value of dk/dr will be kept permanently equal to zero over this range, so 

 that k will be constant, and the equilibrium will be adiabatic. 



Homologous Series 



192. Uniform contraction of the kind considered in 190 may be spoken 

 of as "homologous" contraction, the initial and final configurations being 

 homologous. A series of configurations of equilibrium, each of which may be 

 derived from the preceding by homologous contraction, may be called a homo- 

 logous series. 



On a homologous series, the relations (510), (511) and (512) hold for every 

 pair of configurations. Combining these with* equation (513) we readily 

 obtain the further relation 



k'a'(*-W = ka(*-W (517). 



Let us examine how many of these homologous series there are. Using 

 the relation p = kpi, the equation of equilibrium (509) may be put in the 

 form 



afp-^ = -M q k-Vy. 

 cq 



If M the mass and a the radius are given, and k is given as a function of 

 q, we are able to start from the surface (at which p = 0, q = 1 and M q = M) 

 and determine step by step successive values of p up to the centre q = 0. These 

 values of p, since k is gwen, suffice to determine p and T uniquely. Thus 

 given values of M, a and k determine uniquely one equilibrium configuration. 

 We cannot, by this means, ensure that the total mass obtained by integrating 

 4-TT/or 2 shall be equal to the assumed value of the mass ; this difficulty can be 

 met by admitting as configurations of equilibrium a set of configurations 

 having point-masses, positive or negative, at the centre. With this con- 

 vention, it appears that there are just as many equilibrium configurations as 

 there are sets of values of a and k, k being a function of q. 



