458 Mr. G. A. Schott on Frequencies of Free Vibrations 
possible for the assigned value of j3 ; any smaller value would 
give broader lines and shorter wave-trains (§ 19). 
Since v is the total number of negative electrons in the 
system, whether all be grouped in rings or not, the quotient 
v/n gives the greatest possible number of rings of the system 
which can produce lines of the requisite fineness. This 
number is probably very considerably overestimated, for it 
assumes that nearly all the rings have the same number of 
electrons, while it is nearly certain that many of them have 
many more than the lower limit (n ) for n just found. 
Multiplying v/n by the number of lines given by a single 
ring, we get an upper limit for the number of lines given by 
the system. 
§ 28. In order to see what kind of results can be obtained 
in this way with different assumptions, we shall make the 
calculation for the values v=10, 100, 1000, 10,000, 100,000, 
and for p = 10 -9 , 10 -8 , 10 -7 cm. The constants required are 
e 2 /mC 2 , and mC d /e 2 . For small velocities m=6 . 10" 28 gr., 
e=3-5.10- 10 (E.S.U.), C = 3.10 10 cm. Hence 
e 2 /mC 2 = 2-2. 10" 
13 
-C 8 /g 2 = l-4.10 88 . 
For larger velocities m is greater but a sufficient correction is 
easily applied. 
The method of calculation is best understood from a parti- 
cular case, say j/=1000, p = 10~ 9 . 
Neglecting K we get by (19) /3 2 = 0'22. Thus to a first 
approximation /3 = 0"47, n = 66. 
This is too large for us to neglect the change in mass, 
which amounts to an increase of about 7 per cent, according 
to Kaufmann. The value of Kfor n = 66 is about 48, practi- 
cally 5 per cent, of v. Thus on the whole we must decrease 
/3 2 by 12 per cent., and ft by 6 per cent., getting /3 = 0'44, 
%~59. 
The following table gives the results of the calculation. 
v= 10. 
100. 1000. 10,000. 
100,000. 
p /3. w. >/ . 
0- 
w. 
n . 
/3. ixi. 
n . : /3. 
(i). 
n . 
fi. 
10. 
»„• 
10~ 9 -036 110 11 
•14 
400 
19 
•44 1300 59 ^1 
.^1 
10- 8 -013 j 4-2 8 
046 
14 
11 -15 44 19 -45 
140 
61 
^1 
10- 7 -0043 
013 G 
•014 
0-45 3 -047 1-4 11 -15 
4-4 
19 
•46 
10 
62 
