﻿based on Free Electrons. 



1099 



>of z — z s enter into the calculations. These coefficients are 

 equal tor s and n — s. and for near neighbours vary inversely 

 >as the cube of the smaller of these numbers. We get 

 an approximate account of the mutual action if we take 

 only next neighbours for which s = l. For example, the 

 equation for z 2 with next neighbours £, and z% stands 



fip 2 z 2 + ccz 2 -/3{z 2 -z 1 + z 2 -z. s )=0 "1 

 or ( fjL p* + *-2j3)z 2 + {3(z l + z,) = 0,\. . (38) 

 or say yz 2 + 3i + z z =0, J 



in which 



« = 2 3 /8N = 2>i 3 T 3 /7r 3 N, /3 = n 3 /87r 3 N. . . (39) 



Equations (38) written for each variable constitute a 

 cyclic group, which may be treated by the method of deter- 

 minants or as an equation of finite differences. The latter 

 method gives z n = ( — 1) M (A sin ny + B cos ny) where y = 2 cos 7. 

 The cyclic character is then expressed by the conditions 

 z jl+ i = z l and z n+2 = z 2 , which lor n even require sin ny/2 = 

 •or sin 7 = 0, for n odd cosny/2 = or sin 7 = 0. The deter- 

 minant * itself is given by 



±A n = COSny+{-l) H + l 



For n even the solutions are given by 

 « 7 =7t(2,4, ... 2n); 



for n odd the solutions are given by 



?i7 = 7r(l, 3, ... 2n— 1). 



(40) 



(41) 



The value y = 2 or y = 2ir occurs only with n even ; 

 y——2 or 7 = 7r is common to the two series and involves 

 Z1—Z2 =23=..., which was assumed as basis of the problem 

 in § 25. All other roots occur in pairs which give equal 

 values to y or 2 cos 7; but these equ;il roots of the approxi- 

 mate equation will no doubt be replaced by closely adjacent 

 roots with more exact treatment. It is convenient to refer 

 to the values y= —2 and +2 as extreme values, though the 

 latter only occurs with n even, this extreme for n odd being 



2 COS 77/71-. 



V 



Vi 



1, 



0. 



1 



1, 



y> 



i, 







o, 



1, 



y, 



1 



1, 



0, 



\ 



y 



It would be of interest to get solutions of the 

 determinant with two other neighbours, in 

 which the first row of A B ' would bey, 1, k,0, k, 1 ; 

 where we could suppose &<1. 



4B2 



