of Quasi-permanent Systems of Electrons. 451 
1 +7 . 
Since j3 } 7 are less than unity and 7— J log — — - is negative, 
the form of equations (6) to (9) shows that when we multiply 
a, or a', by a power of 10, for a given value of /3 we di- 
minish n by an amount nearly proportional to the exponent. 
For instance, if in § 5 we assume the breadth of a spectrum- 
line to be 1A.U., in place of '01 A.U., and the number of 
waves in a train to be 100,000 instead of 1,000,000, we must 
multiply a! by 10 3 and diminish n by 10 per cent. 
Physically speaking we see that a ring with a given velocity 
gives finer lines and longer wave-trains the greater the 
number of electrons (of course only on hypothesis B). 
§ 20. We shall here collect together certain results which 
occur in all the theories which we shall discuss. The 
notation is that of Maxwell in his paper on Saturn's Rings ; 
we shall use it throughout for the sake of uniformity. 
Neglecting /3, the force exerted on any one electron of a 
ring of n equidistant electrons by the rest is a repulsion along 
the radius equal to Ke 2 /p 2 ? where 
i=n—l -j 
K=2 
4 sin {in/ny 
2 = 1 
For values of n equal to 10 or more K= —^ — nearly. 
When the ring is subject to a disturbance (q, k) parallel to 
the axis, to the same approximation the force on any one 
electron whose displacement is f is in the direction \ and 
equal to J<? 2 f/p 3 , where 
i=n ~ l sin 2 (km/n) 
<* ~ ^ 4 sin 3 (717/71/ 
i=l 
When it is necessary to specify k we write J& for J. 
Obviously J =0, and J\ — K; the maximum value of J 
n 71 — ~ • 1 
occurs for k= ^, or — — , and for moderately large values 
of n is equal to 0*017 . n 3 nearly. 
When the ring is subject to a disturbance (q, k) in the plane 
of the orbit, with components (f, 77), to the same approxi- 
mation the force in the direction \ is 
and in the direction 77, 
—fil&E/pi + 'Lfrl/p*, 
