RECENT STATISTICAL THEORIES 727 



into the theory of the diffraction of electron-waves, Houston demon- 

 strated that over a wide range of temperatures the resistance of a 

 perfect crystal should vary as the absolute temperature. 



To determine not only the law of variation of resistance with 

 temperature but the actual value of the resistivity for any metal, 

 it would be necessary to evaluate the mean free path of the electrons. 

 By the thoroughgoing corpuscular theory, this depends on the size of 

 the atoms from which the corpuscles rebound; by the wave-theory, 

 it depends on the scattering-power of the individual atom, which thus 

 takes the place of the "size of the atom." The problem of computing 

 the scattering-power of an atom for electron-waves belongs to the 

 new mechanics. Houston was able to obtain good numerical agree- 

 ments for several metals. 



Another way of introducing irregularity into a crystal of an element 

 consists in replacing a small fraction of the atoms, chosen at random 

 here and there on the lattice, by atoms of another element. Certain 

 alloys, known as "solid solutions," are of this type; and it is not 

 only known that the resistance of such an alloy is greater than that 

 of the element which is most abundant in it, but it has been shown 

 by Nordheim that the dependence of resistance on percentage of 

 substituted atoms follows the rule to be expected from the diffraction 

 theory of resistance. ^^ 



Since then the conception of mean-free-path can be reinterpreted 

 in terms of the wave-theory, and since it appears to be possible to 

 deduce from the wave-theory a law of variation of mean-free-path 

 with temperature which can be incorporated intact into the corpuscle- 

 theory it is permissible to return to the corpuscle-picture to set up a 

 theory of conduction of heat and of electricity, and of the thermo- 

 electric effects in crystals. 



We shall apply what I may call the method of the perturbed distri- 

 bution-function, developed by Lorentz. The idea is, to begin by 

 deriving a distribution suiting the actual case. The functions which 

 we have hitherto employed, that of Maxwell and that of Fermi, 

 are isotropic; it is only in the combination (^- + T + f^), hereafter 

 to be called ^'^ that the velocity-components ^, 77, f appear in them.^" 

 These "standard" functions may be appropriate to a uniform metal 

 in which the temperature and the potential are uniform. Evidently 

 they are not appropriate to a metal in which there is an electric held, 



1* The idea that the free paths of electrons extend from one irregularity of the 

 crystal to another was propounded before the advent of the wave-theory of negative 

 electricity. 



1^ I shall use the velocity-components hereafter in lieu of the momentum compo- 

 nents, to conform with the custom. 



