DOUBLE INTEGRALS TO OPTICAL PROBLEMS. 
347 
For this single train, then, 
according to the law 
we have energy falling off from a maximum at k 
sin 2 7 rrn 
n 
o 
0 ) 
(the ordinary law for the diffraction pattern of an edge), where n is the distance from 
the brightest part measured in the scale of reciprocal wave-length. 
Summation for all Molecules having Definite Velocities Loth Athwart and in Line 
of Sight. 
§ 42. For these we have a definite position k of maximum brightness, and definite 
resultant velocity v. We have to integrate for the different lengths of train. 
Now Tait # has shown that, of all atoms moving with velocity v, a fraction e~f p 
penetrates unchecked to distance p, where 
f — 4t tus~ 
We may write this function of v as follows : 
where 
7>- p2 / 1 
-_L -i -L 
T> 1 \ P3 ' 
/ X 9 
= tt-us'' 
= cW) 
P = vh\ 
t( v ) - p +(p 3 + 2 j_ 
9 19 
c~ = 77/^6, 
9 1 I e -VjP 
(ii-)? 
e~ F \lP, 
n = number of atoms in unit volume, 
s = diameter of atom. 
From this we see that, of molecules moving with velocity v, a fraction 
fe~ fp dp 
have free paths between p and p + dp. 
Now, such a molecule will emit an undisturbed train of waves of length between 
V 
r and r + dr, where r — —p, and Y is the velocity of light. 
v f . 
Flence, of all molecules moving with velocity v, a fraction ~e ^ dr, will give free 
paths between r and r + dr. 
* Tait, ‘ Edinb. Trans.,’ vol. 33 p. 72. 
9 v 9 
j-J JL 
