i 4 6 THE QUANTUM [ x . 4 
We now assume that the quantum condition may be expressed 
in the form 
2 j* Tdt — nh io : 40 
where n is an integer. Then we can write 
^HA j* (i 1 + i 2 + i3 + . . . + ijdt = nh . 10 : 41 
assuming that in each part of the integral the integration extends 
over a period corresponding to a particular current. 
This is an arbitrary way of stating the quantum conditions, 
but so is any statement of these conditions, which must be 
regarded rather as a set of rules justified by the results obtained 
than as a consistent and unique theory. 
Now J* i x dt when the integration extends over a correspond 
ing period, will represent the charge, and assuming the charge 
to be built up of electronic units, it must take the form K X e. 
Treating the other terms in the same way, we get 
juHA^i + + . . . + K n )e — nh . . 10 : 42 
The simplest interpretation of this result is to assume, as we 
have done in previous cases, that the sum of the integers is equal 
to n or is some sub-multiple of n. That means that the unit 
tube is defined by (fiH.A)e — h or ^HA = h/e. 
Next consider the electrokinetic energy of the unit quantum 
tube in question. This is given by 
J/iHA(t! + ¿2 + ¿3 + • . . + i n ) 
= ^i K i v i + + . . . + K n v n )e . . 10 : 43 
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since i lt the current in the first circuit, may be expressed in 
the form 
i\ — K x ev x , 
where v t is the frequency in that orbit. 
In most atomic systems it is probable that the integer re 
presented by k is unity in each case, and so the energy would 
reduce to 
lh{y t + v 2 + v 3 + . . . +r n ). . . 10:44 
We see from this expression that the energy of the tube may 
be expressed in the typical form \hv, where v is the frequency, 
which may be the result of adding algebraically a number of 
independent frequencies. We notice that v is a frequency char 
acteristic of the particular quantum tube under consideration.