MAGNETIC TUBES
x. 3]
143
* H. S. Allen, Phil. Mag., vol. 48, p. 429, 1924.
Last of all we have tubes linked with both circuits and the num
ber of these is
N 12 = M 12 (G + i 2 ) . . (2 in Fig. 22)
The total number of tubes is obviously
N t + N 2 + N 12 = Li»! + L 2 f 2 (7 in Fig. 22)
a result which we might have foreseen.
The simple fact, illustrated in Fig. 22, that the various mag
netic tubes associated with two current circuits, may be grouped
together in different ways, is important in connection with the
method of applying the quantum conditions. In his original
work the author * started by classifying the tubes by the latter
method, i.e. by considering the tubes linked only with either
circuit. It is, however, more satisfactory to adopt the former
classification and consider the total number of tubes linked with
each circuit. The reason is that we are then able to apply the
quantum conditions according to the accepted principles in the
way suggested by Professor C. G. Darwin of the University of
Edinburgh. His treatment, with slight alterations in the nota
tion, is given below.
We are dealing with a system depending upon two variables
which in this case might be taken as the two currents i 1 and
i 2 , but we may start by attempting to quantize the general
expression
W = \ax 2 + fxy + |6y 2 . . . 10 : 24
This is a quadratic function of the variables x and y which are
supposed to have the same periodic character, and the cycles
for x and y are supposed to be from 0 to 1. In accordance with
the Hamiltonian method previously discussed we put
Px = ax fy\
Pv^fx + by)'
Making the substitution we find
10 : 25
W = 1 b P* ~ 2 /&A + a Pv
2 ab-f*
10 : 26
Then the motion is determined by the Hamiltonian equations
fyx =
dt
This gives
, . dW ,
etc., x = 7—, etc. .
d Px
p x = constant .
10 : 27
10 : 28
If, then, we apply the quantum condition as given by Wilson
and Sommerfeld (2 :18) we have an integral taken over a cycle
represented by
pp x dx = r Ji, etc., where r x is an integer . 10 : 29
