24 
THE QUANTUM 
[ii. 4 
ma 2 cf>. This angular momentum is taken as the impulse co 
ordinate ft, so that ft = Mci 2 '(f) and is constant, for we are assuming 
a constant speed throughout the motion. Also ft = 
accordance with equation 2 :13. 
The action may be written in the equivalent forms 
1=2 ftTdt — ftftdq 
ÔT 
ôf 
in 
or I 
j* "ma 2 (}) 2 dt = I ma 2 <pdft 
2:19 
v 
Fig. 3.—The Simple Rotator. 
the integration extending over one complete period. Now 
ma 2 ^ is constant and equal to ft, therefore 
Ç2 n r2TT 
I = I ma 2 <j)d(f) — ft I dft = 2nft . . 2:20 
Jo Jo 
The quantum condition is I = rih, so we get 
2 nft = nh 
as given by Nicholson and assumed by Bohr. Further, the 
kinetic energy of the system (which is numerically equal to the 
quantity called W by Bohr) is \rihv. 
Returning to the general case, each pair of variables q k , ft k , 
may be represented by rectangular co-ordinates in a plane, and 
it is a direct consequence of our third hypothesis (p. 22) 
that the condition of the system at any given instant during its 
steady state will be represented by a point on a certain locus