[II. I 
The 
2:8 
j8 THE QUANTUM 
¡uq in a displacement q, so that V = f jliqdq = \fxq 2 . 
total energy is given by 
T+v =Kt) 2+iw2 • • • 
The sum of the two terms on the right is l/ua 2 , or \ma 2 oi 2 , the 
total energy T + V being therefore constant. Since the area A of 
the corresponding ellipse is nab, we find nma 2 co — A so that 
the energy can be written in the form ^ x M or Ar where v 
2/JZ 
is the frequency. 
In Planck’s first theory the only possible stable condition 
of a resonator is one in which its energy is an integral multiple 
of hv. Hence, A the area of the ellipse must be an integral 
multiple of h, so we may write 
A = nh, 
where n is a whole number. Thus according to the present 
theory the phase-plane may be divided up by a series of con 
centric ellipses such that the area between any pair of consecutive 
curves is equal to h, Planck’s constant. The representative 
point of a resonator must always lie on one of the elliptic curves 
which form the boundaries between the regions of equal area 
(Fig. 2). When a resonator either emits or absorbs energy, the 
representative point jumps from one ellipse to another. 
In the case of emission, the point leaps from the ellipse cor 
responding to its stationary state to a smaller ellipse ; in the 
case of absorption the point jumps to a larger ellipse. Con 
sequently both emission and absorption take place in multiples 
of the quantum hv. 
The geometrical mode of representation here employed may 
be compared with the method in statistical dynamics in which 
the co-ordinates determining the position and velocity of a 
molecule are considered as determining the position of a point 
in an imaginary space of the appropriate dimensions. The