354 
MR. C. GODFREY ON THE APPLICATION OF FOURIER’S 
§ 49. The following considerations may tend to remove the doubts which this result 
may arouse. 
The whole set of molecules with given velocities, thwart and in line-of-sight, and 
with given length of free path, will emit light of a certain spectral composition, 
namely, that given by the function " (p. 346), r being the length of the train of 
waves emitted during the single free path. In theory, this will give a set of lines in 
the spectrum of the same pattern as the diffraction lines of a straight edge. Other 
sets of molecules with other velocities and path-lengths will give other sets of lines ; 
the whole aggregate of lines overlapping and compounding to give such spectrum lines 
as actually exist, and were measured by Michelson. 
§ 50. Now, by lowering the pressure of the gas, we may lengthen the average free 
path, and the average trains of waves emitted in a single flight, this lengthening 
being theoretically without limit. The effect of this will be to narrow the curve 
* 9 
Slid - irrn 
——, also without limit. We are tempted, when this happens, to substitute for the 
aggregate of curves ^ ■■ the aggregate of their maximum ordinates. If we do 
this, and also assume that the average length of wave-train is the same for all 
different velocities in the gas, we shall, in effect, be following Lord Rayleigh’s 
procedure, and we shall obtain his expression for the width of the resultant line. 
§ 51. On closer examination, it will be obvious that the molecules moving with 
greater velocities emit, on the average, shorter wave-trains. For, given the velocity, 
the mean free path is l//'(p. 347), a function of v; while the corresponding train of 
waves has length YJvf. It is not difficult to verify that this function of n diminishes 
as v increases. 
g 1 ' 
Now, the molecules with greater velocities in the line of sight have, on the average, 
reater resultant velocities. These, therefore, give shorter trains of waves, and 
smaller ordinates in the energy curve (the maximum of is 77 -T 3 , for n = 0 ). But 
r f\f J 
these molecules will provide light that goes towards the edges of the spectrum line. 
The energy-curve will accordingly be steeper, and the line narrower than would follow 
from the assumption that the mean free path is the same for two groups of molecules 
having; two different velocities. 
§ 52. Furthermore, it is not allowable to substitute for the component curves their 
maximum ordinates, however steep and narrow these curves may become. The 
maximum of is 7 fV 3 . If we substitute these maximum ordinates and then 
form an energy-curve by summing them all into a smooth curve, each will contribute 
sin - vn 
energy proportional to r 3 . But, in reality, the total energy connected with 
+ co 
is 
sin - 7 irn A 
T2 
/i 
dn = tt~t. Thus, trains of length r ought to contribute total energy 
— CO 
