86 Mr. W. Sutherland on the Law of Attraction 



Then the above becomes Apw log — , 



i. e. Apw log n 4- Apw log — '; 



but i\ is so excessively small in comparison with the sen- 

 sible length L that n is negligible in comparison with — • 

 Hence we may write Apia log — " as equivalent to the above. 



Hence for the potential of the finite mass of gas round P 1 

 at the centre of P we can take that of the sphere of matter of 

 radius L, w r hich is, 



AirAp log — . 



Hence for the mutual potential energy of the particle P 

 and the whole mass of gas, we have 



4:7rAmp log - ; 



and for the total potential energy of n molecules, leaving 

 out of count those so near the boundary that a sphere of radius 

 L cannot be described about them so as to lie wholly in the 

 matter under consideration, w r e have, 



2iTAnmp log -, 



changing from the numerical coefficient 4 to 2, because we 

 must not count the mutual potential energy of any tw r o 

 particles twice over. 



If now the mass of gas is allowed to expand (in Thomson 

 and Joule's experiments it expanded by various amounts up 



to six times the original volume), the value of log — remains 



practically constant, and the new value of the total potential 

 energy is 



lirAnrnp' log — , 



where p ! is the density of the gas after expansion. 



Therefore the change of potential energy is proportional to 



M ( P -f), 



M being the mass of gas. 



Now the cooling effect corresponding to this will be obtained 

 by dividing the above by JM.s, where s is the specific heat of 



