Woodioard — The Relations of Internal Pressure, etc. o9 



Since log -7= log( 1 + — — ; — ), it is at once evident that 



for all values of r but little less than r^ (and contraction must 

 begin from r = r^), the work of compression is greater than 

 the work of gravitation ; hence the force of gravitation is 

 insufficient to begin the work of isothermal compression. 

 Much less than sufficient, therefore, would it be to produce 

 isentropic contraction, or any contraction with a rising tem- 

 perature. 



If /'o — r = ds, an infinitesimal contraction, then the two 

 elements of work shown in (18) become 



ds 

 dE^ = 2CT^ 



1 

 



(19) 



ds 

 dE, = SOT^ — 



^0 



' g •" ^ -^0 J. 



r„ 



Equations (19) show that gravity is competent to do but 

 two-thirds of the work required for an initial compression. 

 It is therefore clear that the gas must lose still more heat in 

 order that contraction by gravity may be possible. This loss 

 involves an actual fall of temperature. 



We are thus driven to the conclusion that automatic con- 

 traction is of necessity accompanied by a fall of temperature 

 in its initial stage. Moreover, if steady contraction from one 

 state of equilibrium to another is analogous to the uniform 

 motion of a body under balanced forces, then every interme- 

 diate state is a state of equilibrium, and every element of 

 contraction is an initial one, and the fall of temperature must 

 be continuous. Such is the conclusion that must be drawn 

 from equations (19). 



Now, on the other hand, let us examine again equation (12). 

 This asserts that the mass in a central sphere is proportional 

 to the product of the temperature and radius, i. e., 



2C 

 M = — T r 



Now, if from any cause lohalever contraction of the mass 

 takes place, so that the sphere of radius r^ is reduced in size 



