356 
ME, C. GODFREY ON THE APPLICATION OF FOURIER S 
F(t) = 0 for t — — oo to t = 0 
and F(z) = e~ Kt cos pt for t — 0 to t — + oo . 
(The phase xjj will not make any important difference in the energy-function.) 
jo J o 
7tF(<) = e cos p/3 cos u(/3 — t)cludj3 
_1 
— 2 
1 
2 
CO CO 
dii e * 8 [cos(p + v/3 —- ut) -f- cos (p — u/3 
0 Jo 
du 
k cos ut + p + u sin ut k cos ut — p — u sin ut 
"P (j? U m ) 3 
+ 
tC~ + (j) — uf 
Now k/p is small since the damping is gradual; accordingly, both ■ ^ ^' - + and 
—6 / -ts will be of order 1/p 3 unless p is near to u. In that case, the latter of the 
at + (p — uy 11 r 
two expressions will attain to the order 1/k. We are justified in approximating to 
the extent of neglecting the former expression. 
§ 57. The energy of the train of waves will depend on 
co 
f du 
V 2 + (p-uf 
This function will define the spectrum to which a vast concourse of such damped 
trains is equivalent. 
Now this will be a widened line in the spectrum. The “ half-width ” will be of 
order k in frequency. The half-widths which Michelson has observed for irresoluble 
lines are of order 
lCT 5 X p. 
If 
k is of this order, 
k x 10 s . n . 
- is timte. 
V 
Now 10 5 /p is comparable with the time of 10 5 vibrations. Again, there are on the 
average 10 5 vibrations in the free path. In time t the vibrations are reduced in the 
ratio e -K/ : 1 ; if ut is finite, the reduction of the energy may be very noticeable. 
§ 58. We are thus led to conclude that the vibrations of molecules may be very 
considerably damped in the course of their free paths and yet no widening of lines 
be produced beyond what is actually observed. It will be remembered that the 
kinetic theory, without damping, gave widths varying from one-quarter to two-thirds 
of the widths observed. It seems not impossible that, for small densities, the 
residual width is to be ascribed to radiative damping. When the density becomes 
considerable, the mutual effect of neighbouring molecules will doubtless become so 
important as to obscure both damping and Doppler effect 
