of Quasi-permanent Systems of Electrons. 459 
The unit for a> is 10 16 revolutions per sec. 
The frequency, measured by 27rC/\, in the same unit is 1*9 
for the extreme ultraviolet, 0*24 for the extreme reel, 0*48 for 
the violet. 
§ 29. In studying the frequencies of the lines of the four 
types given in § 25 we must bear in mind that Nagaoka's 
expressions presuppose that v is large ; for values of v less 
than 100 they give only rough approximations to the truth. 
As a matter of fact we are interested mainly in systems 
for which v is large, on account of the difficulty as to 
stability; the small values are merely added for the sake of 
completeness, 
rr , x v 4N 1 -L 1 -2K + 4M 1 
Type (a).— N=« ^ : ' 
By § 20 we find 
4N 1 -L 1 -2K + 4M 1 = 4'03^Log>i-0-300+ ( ^\ 
where Loo- denotes the common logarithm. 
For n=6, 4N 1 -L 1 -2K + 4M 1 = ll-7. 
For n = 60, 4X 1 -L 1 -2K + 4M 1 = 357. 
For n = 450 4N 1 -L 1 -2K+4M 1 =4275. 
°^If we assume that all the rings in each system have dif- 
ferent numbers of electrons, as for instance is the case in 
J. J. Thomson's model, but that the larger rings differ only by 
one electron at a time, then the number of electrons in the 
r(r-\-V) 
largest ring, r, is given by -~^ — - = v. The values are, for 
v = 10, 100, 1000, 10,000, 100,000, 
r= 4, 14, 44, 141, 447, 
where the values for 10, 100 can only be regarded as rough 
approximations, since for small rings the difference from ring 
to ring may exceed one unit. 
These numbers show that the largest ring in every system 
gives a line of type («) in the observable spectrum for some 
value of p between 10~ 8 cm. and 10~ 7 cm. Thus every 
system may give as many lines of this type as it has rings 
with a number of electrons at least equal to n Q for the given 
