RECENT STATISTICAL THEORIES 719 



It is best, perhaps, to regard these formulae as the descriptions of 

 two electron-gases, one composed entirely of magnets parallel to the 

 field, the other of anti-parallel magnets; the actual electron-gas is a 

 mixture of the two. The quantity Z^, -i, for example, is the number 

 of parallel magnets having energy-values shifted downwards through 

 A from the Qs originally-permitted energy-values which lay in the 

 shell s. True, these magnets are no longer themselves in the shell s, 

 owing to the shift; their energies lie between (es — A) and (e^ — A 

 + des), not between e^ and Cs + d^s] but for ease of integration it is 

 better to think of them as being associated with the original unshifted 

 energy-values. The total number of magnets comprised in the 

 parallel gas is then given by the equation: 



2 V r=° (e)^'~de 



^^ = (^ p i'^-'nr' I ,JJ.)..Vi • (84) 



For brevity denote by L the constant before the integral; and use the 

 symbol for the function defined as follows: 



X°° (f)^'^de 

 ^■^^l- (85) 



Then expanding iV"i as a power-series in the variable A/kT which is 

 fioH/kT, we find: 



Ni = L 



ct)(B) ■p^4)'(B) -f- terms of higher order in H 



(86) 



Similarly one obtains, for the total number N2 of electron-magnets in 

 the "anti-parallel gas," a formula: 



N. = L 



(t>{B) + ^^ (i>'{B) + terms of higher order in // 



(87) 



Now the total magnetic moment of the "parallel gas" is A^i/xo in 

 the same sense as the field, and the total magnetic moment of the 

 "anti-parallel gas" is N^hq in the sense opposed to the field; so that 

 the net magnetic moment of the entire assemblage of electrons is 

 (TVi — NiJiJ-o- We will carry out the computations only for values 

 of H so low that we may ignore all terms beyond the second in the 

 expansions for Ni and A^2- The net magnetic moment is then approxi- 

 mately proportional to //; its quotient by //, the susceptibility x of 

 the electron-gas, is constant. It is a fact of experience that with 

 nearly all paramagnetic substances the susceptibility -is independent 

 of H up to the highest attainable values of this variable. The limi- 



