212 Mr. G. A. Schott on the Electron 



diminish with increasing index, and are small compared 



with — 2 . The equation is transcendental, and thus has an 



infinite number of roots; for stability all must be real. We 

 shall, however, not concern ourselves with the question of 

 stability. 



When — is small enough we may neglect the higher 



powers in the series ; we then obtain a quadratic, which is 

 not very different from Nagaoka's equation, and has two roots 

 of the same order of magnitude. For values ofk = 0, + 1, +2, 

 these correspond to observable spectrum-lines, six in number 

 at most. o 



§ 21. When — is large, the equation reduces approximately 



to 



CJmp ^ gft j (qP\* 



— 2^ +« 3 — +a 4 ( ) + 



er co \ ft) / 



This equation has an infinite number of large roots. Can 

 any of these give rise to observable spectrum-lines ? 



In order that they may do so the frequencies must fall 



within the proper limits, that is, ^~ + kft. which is equal to 

 t) o) ' 



-^, must lie between -0063 and -0008. We find 



as= C s l cot {2h ~i~ 1>r [2j * +i{ (2 * + m 



+ (1 -/3 2 )0 + l)/?J' 2s+ i (2* + 1)/3}] , 

 £(3+ff ) w 

 ■6(1-/S 2 ) °°V 



This is exceedingly small compared with — - 2 — ; hence 



^- is a very large number ; it follows that ~ + kft cannot 

 o) n a) 



be small, for k is less than + -, and ft is less than unity 



The only exception might occur for n verv large and ft nearly 

 unity. For n = 1000, ft = • 9, we find 



Og< 1800, 



&) 



>26. 



In this case the distance between consecutive electrons is 

 only 50 times the radius of an electron. Even when the 



