28 SECTIONAL ADDRESSES. 
number of orbits with the same major axis but with different eccen- 
tricities, while all were characterised by the same energy value. For 
each value of n the number of such orbits was given by the number of 
ways in which n could be made equal to the sum of two integers, 
including zero. For example, if n were equal to 1 only a single orbit 
could exist. If were equal to 2, then since 2=2+0 and 2=1+1 we 
could have two orbits. If n were equal to 3, we see again, since 3=3+0 
or 3=2+1 or 8=1+2, that we could have three orbits, &. For 
each value of n there could exist a definite number of equivalent orbits. 
If we put n=n,+n, it can be readily shown that the eccentricities of 
these equivalent orbits are given by 
2 2 
0 Ra Tr al Ny 
pred nm (m+ 10)” ; ‘ é ; (2) 
If 2b be taken to represent the minor axis of the different equivalent 
elliptical orbits, it follows that the ratio of the semi-axes is given by 
SI nae cad scr jb ih3) 
FiaG@. 1.—H Orbits. 
Illustrations of such equivalent orbits for the hydrogen atom with differ- 
ing values of n are shown in Fig I. On this view the Lymans spectral series 
v=K(1-4,) originates in transitions to the n=1 orbit, the Balmer 
™m 
series v=K(5—4)in transitions to either of the n=2 orbits, and the 
2° mM 
Paschen series v=K(5- *,) in transitions to one or other of the orbits 
m 
of the n=8 group. 
