Woodward — The Relations of Internal Pressure, etc. 57 



This result coiiUl hardly have been anticipated. Another 

 deduction will soon be made from Eq. 12. 



11. 



Can the above mass of gas contract automatically through 

 loss of heat, and be again at rest? If so, what vjill be the 

 effect upon temperature? 



It is not easy to see how such an isolated mass filling all 

 space could lose its energy in the shape of radiant heat; but 

 for the sake of the discussion, we may assume it. 



In the process of contraction all particles move directly 

 towards the center, the amount of motion being in all cases 

 proportional to the radius. This follows from the law of 

 distribution of the mass as shown by equation (10) the 

 temperature being uniform, though not necessarily constant. 

 All spherical shells of the same mass will still be of the same 

 thickness, and in all cases the new volume of a unit mass of 

 the gas will be proportional to the cube of its new radius; 

 that is, from purely geometrical reasoning (see figure), 



V r ^ 



As the force of gravity is the only force to produce con- 

 traction (i. e., do the work of compressing the gas), let us 

 compare its capacity for doing work with the work of com- 

 pression required, with a view to finding what change of 

 temperature may be necessary. 



Let us suppose that the entire mass has contracted, and 

 that within it the sphere with radius r^ has contracted to the 

 sphere whose radius is r. Its mass has of course remained 

 constantly M^. The attraction (i. e., the weight) of a unit 

 mass on the surface of this inner sphere is in general during 

 the contraction 



TcM, 



and the energy exerted by gravity in moving it through a 

 distance — dr is 



kM^dr ^ 



