446 Prof. H. Nagaoka: Kineties of a System of 
positively charged particle. The investigations on cathode 
rays and radioactivity have shown that such a system is con- 
ceivable as an ideal atom. In his lectures on electr ons, Sir 
Oliver Lodge cails attention to a Saturnian system which 
probably will be of the same arrangement as that above 
spoken of. The objection to sucha s system of electrons is that 
the system must ultimately come to rest, in consequence of 
the exhaustion of energy by radiation, if the loss be not 
properly compensated. 
To begin with, it is necessary to show that the system 
is stable. Denoting the distance of particle & from the 
centre of mass by R;, the total energy of the system by U, 
and the potential energy by V, we can easily prove that 
a p2_fh 5 
dt? mRi=4U =2Y, s . = < (1)* 
2 
for the law of force varying as the inverse square of distance. 
In order that ym? may remain finite, it is necessary that 
2U—V should eller be always positive nor always negative, 
but assume oscillating values. Let the distance between the 
2 
e 
particles /: and J be v,, the repulsion between them —, the 
rel 
radius vector of the disturbed circular orbit 7;, and the 
: ; e 
attraction between k and the central particle —, the angular 
aye —_ 
k 
velocity about the centre ;, then es pai 
2U—V= meri + +355 - = i = 
Kl eo 
Remembering that mo,rZ=yx Daath Fe angular mo- 
mentum, which remains constant, we obtain 
20 Vey OE Se 
a Vk TE 
Thus 2U--V_ will assume oscillating values, when 7; and 
rj are subject to small disturbanees, provided the quantities 
e, E, and wy; and the mean palin ot +, 7, are such that 
2U —V assumes oscillating values, sometimes exceeding and 
sometimes falling short te zero. We notice at once that 1D 
must be very larg ge compared to e. 
The small oscillations of a particle may be radial or normal 
) the plane of the orbit, and those corresponding to the dis- 
ne ance of 1,, will give rise to condensation and rarefaction 
of the particles arranged in a ring. The oscillations of 
* Jacobi, Vorlesungen tiber Dynamik, Werke, Supplementband, p. 29. 
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