﻿based on Free Electrons, 



1103 



positive centre is possible, that is for n>4, the last bracket 

 in (48 a) is positive, and the two values of q 2 real and 

 positive. If the oscillation can be identified, this is a means 

 of determining yu,, which can be used with any value of 

 n>4 by calculation of cf>. 



The asymptotic value is ??i 1 /m 2 = 143'8v 2 / n V 2 ; the roots 

 are N</ 2 = 2$/^ = 2/< 3 T 3 //x7r ;? and Nq' x ~ 3;/f in asymptotic form, 

 or with our previous value of fi, q = 16'72n and q' = ±'112jn. 

 The number q' corresponds to the new ionic period, and q is 

 not altered, to this order in ?i, from the value previously 

 found for the sum of displacements. 



For a negative unit at the centre the equations are changed 



to m^o^nazx — nftz 2 , m 1 ( < 2 + £i)==—«3i— 7(^1 —#2)5 



m 2 (z 4- i 2 ) = /3z 2 + ?Oi— z 2 ) ; 

 and elimination as carried out above leads to 

 /xYN' 2 -^7 2 N{2.i' 3 (/) + n- (n + l) t ^ 3 } 



->^ 3 {2</>(^ 3 -l) + l} = 0. (48 6) 



One of the roots is necessarily negative when #>1, and 

 on examination it appears that this is also true for n— 2, 3, 

 or 4, cases for which x< 1, that is 2(/)(a i3 — 1) + 1 is a positive 

 quantity. By the criterion of axial stability, therefore, the 

 admission of a positive unit centre and the rejection of a 

 negative unit are justified. 



§ 34. The equations (45) may also be applied to test the 

 cohesion of the system under external force in the direction 

 of the axis. Stability is a part of such a test, but we may 

 take a further step and inquire into the extent of internal 

 displacements when the various elements move with a 

 common acceleration f. Thus we write z =f, z 1 = = z 2 , 

 and examine the magnitude of Z\Z% for a given external 

 force Z. 



Multiplying the second and third by n and adding all 

 equations, we get 



Z = {(w-kl)»i 1 + »2 2 }/, or Z — ?Hi/=nM/, 

 Z + w 2 /=(ti+1)M/ where M=m L + m 2 . 

 The three consistent equations are then 



M/= az, - Zft nMf= y(z x -z 2 ) - ccz u 



(n + l)Mf=y(z l -z 2 )-/3z 2 , 

 of which the solution is )> . (49) 



{7(«-/3) + «/3}m = (7-^)M/, 

 {y(*-/3) + *j3}z 2 ={y-(n + l)u\Mf. 



