ENSEMBLE OF SYSTEMS. 77 



We have therefore 



< 226 > 

 " < 227 > 



* (228) 



The average values of the powers of the anomalies of the 

 energies are perhaps most easily found as follows. We have 

 identically, since e is a function of , while e is a function of 

 the jt?'s and <?'s, 



all f 



phases 



J. . . J[ e(e _ i) _ h (e _ ;) 2 * J e~0d Pl , ...dy. 



(229) 

 _ i_ x enyj 



phases 



i. e., by (108), 



(230) 



* In the case discussed in the note on page 54 we may easily get 



which, with e g 6 , 



gives 



rr^j = Qe + e^) (,-*j = |Qe + * 



Hence c e a ft = c*. 



Again (e - 6 ) = e - e a + 2 ^ (e - ea)*- 1 , 



which with e e = n & 



gives 



(e - )* = (n 6 + 02^) (e - ea)*- 1 = n (w + 02^)*~ J 0, 



hence {7^j = ?^ + *> e. 



