34(5 
MR, C. GODFREY ON THE APPLICATION OF FOURIER’S 
from the whole gas will consist of a great number of finite trains superposed. We 
must consider those trains as practically independent. It is true that each indi¬ 
vidual train is connected in one respect, namely, instant of beginning or ending, with 
two others, the trains emitted by the same molecule before and after. But this 
element of regularity will be overwhelmed by the independence of the different 
molecules. All we have to do is, to find the Fourier integral equivalent to a finite 
train of waves, to find the distribution of energy in the scale of frequency, and to 
sum up the energy for all possible trains. 
Fourier Analysis of a Train of m Complete Sine Waves. 
§41. The general theorem is 
x +a> 
77 f(x) = j ( COS a>(\ — x)f{f)d(o tlX. 
Jo J 
In the present case f(x) = 0, except from 0 to 2wm/K , within which limits f(x) = 
cos KX. 
Thus 
CO — 
7 rf{x) = cos 6 j(X — x) cos k\ dco d\ 
r\ J r\ 
0 JO 
2iTin Ik 
~ -2 {cos ( w + K )^ — wX “h cos (a ) — k)\ — cox]dco dX 
’ Jo Jo 
'0 J 0 
00 
f • 
— I dco 
l, 
7 Tin\ . 7 Tlllco 
cos ©I x — —j sin — 
cos oo \x — 
00 T fc 
+ 
7 i m 
irvioo 
Sill 
O) — K 
■ 7 rvuo ■ TT/nco 
SUl- Sill- 
If we consider the quantities-— , —— , we see that the latter attains to a 
1 00 + K 00 — K ' 
maximum value ttw/k at oo = k, and that the former is small in comparison since m 
is generally a considerable number. 
We shall be concerned only with the values of co near to k ; accordingly the first 
term shall be neglected. We shall then have a distribution of energy, 
sin 2 mmol k 
(ffl - K? ’ 
needectinef numerical coefficients which do not alter the distribution. 
Let oo — k — 2irn ; then at a distance n from the maximum (n being reciprocal 
wave-length) we have energy proportional to 
f d co 
J o 
. 0 7 TTTb. 
Sill-- (co 
K 
- *) 
sin "irrn 
n~ 
n~ 
where r is the length of the train. 
