172 Physical Properties [CH. vu 



By division we obtain 



.(415), 



. 

 2 dV 



n + E^T 

 the quantity 7 denoting, as before, the ratio of the two specific heats. 



203. In the case of hard elastic spheres, we have F=0 and n = 3, since 

 the only energy of the spheres is their energy of translation, which is 

 represented by three momentoids. Hence for a gas composed of such 

 spheres, 



7 = lf ................................. (416). 



This is (except for a possible small error) the value of 7 which is observed 

 for mercury, argon, and as far as is known, for all monatomic gases. We 

 must not, however, infer that the molecules of these gases are hard elastic 

 spheres. Each of these gases, when raised to incandescence, exhibits a 

 spectrum consisting of thousands of distinct lines, and therefore is pre- 

 sumably capable of executing vibrations of thousands of distinct periods. 

 If, as we believe, all the molecules of the gas are exactly similar, each 

 molecule must be capable separately of executing vibrations of all these 

 periods. In other words the kinetic and potential energies of a molecule, 

 specified as a dynamical system through coordinates fa, < 2 <, must 

 contain thousands of terms of the forms 



2L = a^ + a 2 < 2 2 + ... + a s <j> g 2 ..................... (417), 



and since the evidence of the spectrum shews that the vibrations of any 

 single type are (very approximately) isochronous, the coefficients c^...^, 

 &J...&3 must be constants. Averaged throughout the motion we find, as 

 usual, that the mean kinetic energy of any vibration is equal to the mean 

 potential energy, and hence, throughout the gas, 



The quantity a^ 2 is, however, double the mean energy of a momentoid. 

 Hence each line in the spectrum of a gas denotes a contribution to the 

 mean energy of each molecule, equal to double the mean energy of a 

 momentoid. 



From this it appears that the value of n in equation (415), must be 

 taken to be some thousands, and therefore that the value of 7 given by 

 equation (415) would, so far as experimental observation goes, be in- 

 distinguishable from unity. 



It has, however, already been mentioned that the value observed for 7 in 

 the case of the monatomic gases is very approximately equal to If. A table 

 will be given later shewing the observed values of 7 for a large number of 



