137.

The principal difference, compared with the calculations for an ideal gas in the preceding chapter, lies in the fact that we do not now assume the distribution densities $w_{1}$ , $w_{2}$ , $w_{3} \dots$ of the oscillators among the separate region elements to vary but little from region to region as was assumed in [sect:129.] Sec. 129.

Accordingly the $w$ 's are not small, but finite proper fractions, and the summation over the region elements cannot be written as an integration.

In the first place, as regards the shape of the region elements, the fact that in the case of undisturbed vibrations of an oscillator the phase is always changing, whereas the amplitude remains constant, leads to the conclusion that, for the macroscopic state of the oscillators, the amplitudes only, not the phases, must be considered, or in other words the region elements in the $f ψ$ plane are bounded by the curves $C = const .$ , that is, by ellipses, since from [eqn:(207)] (207) and [eqn:(209)] (209) ${(\frac{f}{C})}^{2} + {(\frac{ψ}{2 π ν L C})}^{2} = 1 . \Label [e q n] (211) \upshape (211)$

The semi-axes of such an ellipse are:

$a = C and b = 2 π ν L C . \Label [e q n] (212) \upshape (212)$

Accordingly the region elements $1$ , $2$ , $3, \dots$ $n \dots$ are the concentric, similar, and similarly situated elliptic rings, which are determined by the increasing values of $C$ :

$0, C_{1}, C_{2}, C_{3}, \dots C_{n - 1}, C_{n} \dots . \Label [e q n] (213) \upshape (213)$

plus 0.75em minus 0.25em

The $n$ th region element is that which is bounded by the ellipses $C = C_{n - 1}$ and $C = C_{n}$ . The first region element is the full ellipse $C_{1}$ . All these rings have the same area $h$ , which is found by subtracting the area of the full ellipse $C_{n - 1}$ from that of the full ellipse $C_{n}$ ; hence

$h = (a_{n} b_{n} - a_{n - 1} b_{n - 1}) π$

or, according to [eqn:(212)] (212),

$h = (C_{n}^{2} - C_{n - 1}^{2}) 2 π^{2} ν L,$

where $n = 1$ , $2$ , $3, \dots$ .

From the additional fact that $C_{0} = 0$ , it follows that: $C_{n}^{2} = \frac{n h}{2 π^{2} ν L} . \Label [e q n] (214) \upshape (214)$ Thus the semi-axes of the bounding ellipses are in the ratio of the square roots of the integral numbers.

158