4G0 Mr. G. A. Schott on Frequencies of Free Vibrations 
system. Hence the largest possible number of lines of type (a) 
are, for 
v= 10, 100, 1000, 10,000, 100,000, 
1, 7, 31, 123, 38G, 
on the supposition (1) that the number of electrons is different, 
but only by one unit, from ring to ring. 
On the very much less likely supposition (2) that the rings 
all have the same number of electrons, and that the least 
possible (» ), we get absolutely the largest possible number 
of lines, namely, for 
v= 10, 100, 1000, 10,000, 100,000, 
1, 12, • 90, 526, lb'12. 
§ 30. Type {!>). — N = gj nearly. 
The table (§ 28) shows that these lines only fall within the 
observable spectrum for systems for which v< 1800, for larger 
systems they lie beyond the extreme ultraviolet. 
Type (c). — R = 2(o nearly. 
These lines fall within the observable spectrum when v< 470, 
otherwise they lie beyond the ultraviolet. 
Type (d). — N = 3&> nearly. 
These lines fall within the observable spectrum when v< 210, 
otherwise they lie beyond the ultraviolet. 
"We must also remember that lines of type (d) are very 
weak and can only be seen in exceptional circumstances. 
§ 31. Collecting our results together, we conclude that the 
number of lines to be expected is probably less than the value 
(1), and certainly less than the value (2), given in the following 
table : — 
v .=10, 100, 1000, 10,000, 100,000. 
Xo. of strong lines (1)= 3, 21, 68, 123, 386. 
„ weak „ (1)= 1. 7. 0, 0, 0. 
„ strong „ (2)= 3. 36. 180, 526, 1612. 
„ weak „ (2)= 1, 12. 0, 0, 0. 
All these are estimated on the assumption that 10~ 7 cm. is 
the maximum allowable radius, even for the largest rings. 
A smaller maximum materially reduces the values given ; in 
fact a maximum 10 -8 cm. gives no lines whatever for any 
system (§ 29). 
§ 32. In order to identify these systems with atoms we must 
calculate their masses and compare them with the atomic 
weights, bearing in mind that the mass of the hydrogen atom 
is equal to that of 1700 negative electrons. 
