444 Mr. G. A. Schott on Frequencies of Free Vibrations 
radiation from its atoms is underestimated, the argument o£ 
§ 9 is rendered all the more cogent. 
§ 11. "We shall now examine the frequency equations of a 
ring of the qua si -permanent system of § 9, in order to form 
an estimate of the frequencies to be expected from a system 
whose scale is comparable with that of an atom. Estimates 
of this kind are frequently omitted on the ground that they 
are illusory, because the dimensions of an atom are not 
sufficiently well known. However, in the present state of 
knowledge tbis criticism is not quite justifiable ; and besides 
we merely require superior limits for our purpose. Such 
quantities as the number of molecules in 1 c.c. of gas under 
normal conditions (4 . 10 19 ), the mass of the atom of hydrogen 
(10~- 4 gr.), and the diameter of a molecule (at most from 
10" 8 to 10 -7 cm., probably nearer the lower limit), have been 
calculated by various methods with consistent results, and 
are sufficiently well established to make estimates of fre- 
quencies and wave-lengths of considerable use in judging the 
merits of a proposed model of an atom. We should have 
considerable hesitation in accepting as a working model a 
system whose wave-lengths were very different from those of 
light- waves. 
§ 12. This is the proper place to notice a fundamental differ- 
ence between systems of electrons in orbital motion and systems 
built up of Hertzian vibrators or of elastic bodies. Systems of 
the latter types of different linear dimensions, but otherwise 
similar, have their wave-lengths in the ratio of their linear 
dimensions. Now, speaking roughly, wave-lengths of light- 
waves are a thousand times as great as the linear dimensions 
of the atom ; therefore the wave-lengths of the free vibrations 
of the Hertzian, or of the elastic, system must be roughly one 
thousand times its linear dimensions, in order to furnish 
a satisfactory atomic model. Obviously simple vibrators, 
spheres, ellipsoids, rods and the like, do not satisfy this con- 
dition, so that special assumptions are necessary; for example, 
a suitable Hertzian vibrator might consist of two conductors 
so close together as to have a capacity one million times the 
linear dimensions of the system. 
On the other hand, a ring of electrons involves three linear 
quantities in its specification, namely the radius of the ring, 
and the radii of the negative and positive electrons, of which 
the two latter remain the same when the linear dimensions of 
the system are altered. For this reason the wave-lengths 
of the free vibrations of the ring are in no direct relation to 
the radius of the ring ; in fact it may happen that a very 
small ring emits longer waves than a large one. The size of 
