192 The Evolution of Gaseous Masses [CH. vm 



surface. In a uniform shrinkage such as we considered in the last section, 

 each element of the gas will retain the same value of q throughout the 

 shrinkage. Changing the variable from r to q, equation (508) becomes 



(509), 



where M g is the mass inside a sphere of radius raq\ this of course 

 remains unaltered throughout shrinkage. 



Now let the mass shrink uniformly in a ratio to a new configuration of 

 radius a', so that of = aO. Let p', p, T be the new values of p, p, T. Since 

 the shrinkage is supposed uniform, the density p at each point will be I/O 9 

 times the old density p, so that 



p 'a'* = p a 3 .............................. (510). 



Multiplying both sides of equation (509) by a 3 , it appears that the new 

 configuration will be one of equilibrium if 



'4 &!_.&. 



dq dq 



at every point. Integrating with respect to </, we have as the condition of 

 equilibrium 



a'*p' = a 4 p .............................. (511). 



Dividing by corresponding sides of equation (510) and comparing with 

 equation (507) we find 



a'T' = aT .............................. (512). 



Thus if a spherical mass of gas shrinks uniformly from an equilibrium 

 configuration, the new configuration will also be one of equilibrium provided the 

 temperature at every point is made to vary inversely as the radius of the 

 sphere. 



This is commonly called Lane's law*. The analysis has not shewn that 

 the natural flow of heat will be such that a uniform shrinkage will take 

 place it merely shews that if, for any reason, a uniform shrinkage does occur, 

 and the new configuration is one of equilibrium, then relation (512) must be 

 satisfied at every point of the mass. 



It must be noticed that in this law T js the temperature of a given internal 

 element of the gas, corresponding to an assigned value of q. The emission of 

 radiation from the mass comes, not from a single layer corresponding to a 

 single value of q, but from a number of layers near to the surface. Thus the 

 law (512) has no application to the temperature of a star as determined from 

 its emission of radiation this is a different question altogether, to which 

 we shall turn our attention later. 



* J. Homer Lane, " On the Theoretical Temperature of the Sun." Ainer. Journ. Sci. 53 (1870), 

 p. 57. 



