of Quasi-permanent Systems of Electrons. 457 
These results show clearly that a single ring cannot pos- 
sibly account for spectrum series, and that for two reasons : 
(1) The absolute frequencies (N) are approximately in 
arithmetical progression. 
(2) The number of lines observable is far too small, because, 
as I have proved elsewhere, the intensities of successive lines 
after the first two or three diminish far too rapidly. 
§ 27. We must now examine whether a system of ring 
can account for spectra, each ring contributing one or more 
lines. This view it is true has difficulties ; for the similarity 
in the structure of lines of the same series, and in their 
behaviour in a magnetic field and under pressure, is frequently 
assumed to indicate that they are produced by the same 
vibrator. But seeing that all rings of the same system are 
necessarily linked together, perhaps all that we need assume 
is that they are produced by vibrations of the same type, 
though of different rings. Thus it becomes necessary to 
investigate under what conditions the frequencies (N) just 
found fall within the limits of the spectrum. For this 
purpose it is necessary to estimate the value of co. 
co is given by equation (10) of § 23. Introducing the 
quantity /3 = Gop/C, we get 
(0=7 r^-2 (lb) 
and 
P = ^— K)<?2 (19) 
We must remember that the table of § 19 gives an upper 
limit to the value of /3 for each value of n. Thus when n, v 
are given, equations (18), (19) give respectively upper and 
lower limits for co, p, which must not be transgressed if the 
system is to give fine spectrum-lines. 
Again, an upper limit for p is given by the condition that 
the ring must be small compared with the whole system (§ 22) ; 
thus p cannot be as large as the radius of the atom which the 
system represents. With this upper limit for p equation (19) 
gives a loicer limit for /3, and equation (18) a lower limit 
for co, quite independently of the condition for fineness of the 
spectrum-lines. 
This condition, however, with the table of § 19 gives a 
lower limit (n ) for n, This is obvious on hypothesis (A), 
where n is determined by j3 by the table. On hypothesis (B) 
we saw that to produce lines of a given fineness and degree of 
homogeneity, the value of n given by the table is the least 
