348 
MR. C. GODFREY ON THE APPLICATION OF FOURIER’S 
Returning to the expression for the energy of a single train of length r (i.), we see 
that with the aggregates of molecules now under consideration (definite thwart and 
line-of-sight velocities) we have for n a proportion of energy 
fv f -fr ■ -■ 
— e v sm 
n~ V Jn 
TTftr . dr 
fv 
2n 2 V . 
e y r (l — cos 277/z?’) dr 
fv 
1 V 
vf 0 
— —COS I'KTir + 
Y 
27 m sin 277 nr\ 
L-f 
2 n*Y 
0 
* 
+ 
_ 
4t r-rd j 
I 
(iii.) 
We next integrate for a definite velocity p in the line of sight, and all possible 
velocities q athwart. 
The proportion of molecules with thwart velocities between q and q + dq is 
qe~ h<1 \lq. Hence, omitting the \ from (iii.) (it does not affect the distribution of 
energy), we have 
(iv.) 
where if = pr + q : . 
Lastly, we introduce all possible velocities in line of sight. 
Here the Doppler effect enters ; the mid-point of the spectrum (iv.) will be 
different for different p’s. Let x be the distance from the centre of the final spectrum 
line (measured, as before, in reciprocal wave-lengths), we have 
+ 00 
— 00 
1 
dp dq . 
27tY/ 
(v.) 
This integral, regarded as a function of x, gives the distribution of light in the 
spectrum. 
To make further progress, we will change the variables of integration from p and 
q to q) and v, where v 2 = jr + (f. We must remember that /is a function of v. 
