﻿based on Free Electrons. 1077 



Two solutions are possible : (i.) when the value assigned to 

 f(x) is greater than n and less than the maximum, and 

 (ii.) when the assigned value is negative and greater than 

 the second minimum. Case (i.) requiring c n >2n can only 

 occur if n>473, a number outside the range considered. 

 Case (ii.) does not occur with vacant centre ; but occurs 

 once, viz. with n = 4 for a positive centre. Here 

 ±d = *25+'707 = *957, and the assigned value of f(x) is 

 — •043, lying between and the minimum —'059, which 

 occurs near a?=2*39. The second solution, on the up grade 

 of the curve, corresponds to a position of instability. 



The comparative straightness of the graph near x = l 

 suggests a method of approximation to the value of f(x) 

 when n is great and x— 1 small, which gives at once the 

 asymptotic value for n great and a serviceable approxima- 

 tion for moderate values of n. The solution of (6) by 

 numerical test needed for small values of n becomes 

 laborious between 10 and 20, and for prolonging tables 

 beyond this range the use of such a method as is proposed 

 is indispensable. 



§ 11. With x — l = f a small quantity and 6 r = (2r+l)7r/n > 

 the general term of f(x), viz. 



(1-x cos r ){l + x 2 -2x cos 6 r ] -*i\ 

 may be written 



{.<l-cos r )-f}{2a(l-cos r ) + f }~W 

 and expanded in terms of f, as 

 f(x) = ii"- 1 /2(2-2 cos 6 ¥ )-V*-tar*fl(2-2coa e r )~ 3 ' 2 

 - ffar 3 < 2 (2-2 cos 6> r )- 3/2 + |^" 5/2 (2-2 cos r y 

 Thus if 



•3/2 



2, = Scosec(2r + l)7r/2n, % 3 ="i cosec 3 (2r+l)7r/2n ..., 



• • • (11) 

 fix) = i«-«%-ie»-*%-AC , »-*%+A«%- • (i2«> 



When the general term of fl - J, viz. 



x\x - cos r ) { 1 + x 2 - 2x cos 6 r ] - 3 2 , 



