of Quasi-permanent Systems of Electrons. 443 
For generally the coefficients of high harmonics in (3) are 
small provided the rings be not close together ; and it is 
sufficient if either s or s' be a large number when the rings 
are fairly near together, smaller when far apart. This con- 
dition can always be satisfied, except for special values of n, n' . 
§ 9. When the electron belongs to a ring which has not 
the same axis as the first, the coordinates r and 0, as well 
as (j>, involve t, and so also do the coefficients of the circular 
functions in (3). Expanding them in Fourier's series we see 
that the mechanical force on the moving electron now involves 
harmonics of all integral orders, and not merely of the orders 
n, 2n, 3??, .... Consequently it is no longer possible by a 
suitable choice of the values of n, n 1 ', to ensure that the 
amplitudes are very small for the dangerous vibrations for 
which m = 0, +1. Hence considerable radiation occurs, and 
the system is not permanent. A fortiori the same thing 
occurs when the rings are not circular and are oriented in an 
arbitrary manner. 
We conclude that no system of electrons can be permanent 
unless its electrons be grouped in circular rings of equidistant 
electrons, all rotating about the same axis. 
§ 10. This conclusion can be at once extended to a system, 
which is not solitary, as we have hitherto supposed, but is 
surrounded by a large number of other systems, just as an 
atom in a radiating gas is surrounded by a very large number 
of other atoms. It is true that the energy emitted by the 
system, or atom, in this case is, at any rate in part, replaced 
bv energy absorbed by it from the field due to the surrounding- 
systems, or atoms, and that to an amount depending on the 
absorptive index of the gas for its own radiations. But the 
observations on the degree of homogeneity and the fineness 
of spectrum-lines, on which the conclusion of § 9 has been 
based, themselves apply to a complex of atoms, and not to a 
solitary atom. 
In the same way measurements, such as those of E. Wiede- 
mann, of the amount of energy radiated per second from a 
flame or other source, enable us to calculate the net loss of 
energy of an atom due to radiation, that is, the excess of the 
amount emitted above the amount absorbed. If the absorption 
be considerable and be neglected, the net loss is still correctly 
estimated on the average; but since in the case of large 
absorption the energy actually radiated by the flame comes 
from a thin surface layer only, the amount of energy emitted 
from a single atom, and the net loss of each surface atom, 
are both underestimated. Since in this case the observed 
spectrum-lines are due to the surface layer alone, and the 
