CHAPTERS ON THE STARS. 141 



increase from the surface to the center. The law of this increase is 

 incapable of accurate expression, but is not necessary for our present 

 purpose. 



Let the inner circle, C D, represent a spherical shell, situated any- 

 where in the interior of the mass, but concentric with it. Let E P be 

 the corresponding shell after the contraction has taken place. The case 

 will then be as follows: 



The two shells will by hypothesis have the same quantity of matter, 

 both in their own substance and throughout their interior. 



In case B the central attraction being as the inverse square from 

 the center, will be four times as great for each unit of matter in the 

 shell. 



This force of attraction, tending to compress the shell, is, in case 

 B, exerted on a surface one quarter as great, because the surface of a 

 shell is proportional to the square of its diameter. 



Hence the hydrostatic pressure per unit of surface is 16 times 

 as great in case B as in case A. 



The elastic force of a gas, if the two bodies were at the same tem- 

 perature, would be 8 times as great in case B as in case A, being in- 

 versely as the volume. 



The hydrostatic pressure being 16 times as great, while the elastic 

 force to counterbalance it is only 8 times as great, no equilibrium would 

 be possible. To make it possible, the absolute temperature of the gas 

 must be doubled, in order that the elastic force shall balance the 

 pressure. 



That a mass can become hotter through cooling, may, at first sight, 

 seem paradoxical. We shall, therefore, cite a result which is strictly 

 analogous. If the motion of a comet is hindered by a resisting medium, 

 the comet will continually move faster. The reason of this is that the 

 first effect of the medium is to diminish the velocity of the object. 

 Through this diminution of velocity, the comet falls towards the Sun. 

 The increase of velocity caused by the fall more than counterbalances 

 the diminution produced by the resistance. The result is that the comet 

 takes up a more and more rapid motion, as it gradually approaches the 

 Sun, in consequence of the resistance it suffers. In the same way, when 

 a gaseous celestial body cools, the fall of its mass towards the center 

 changes from a potential to an actual form an amount of energy greater 

 than that radiated away. 



The critical reader will see a weak point in this reasoning, which it 

 is necessary to consider. What we have really shown is that if the mass, 

 assumed to be in a state of equilibrium when it has the size A, has to 

 remain in equilibrium when it has the size B, then its temperature 

 must be doubled. But we have not proved that its temperature actually 

 will be doubled by the fall. In fact, it cannot be doubled unless the 



