ABSORPTION OF LIGfIT IN GASEOUS MEDIA. 
387 
or 
The first term in the brackets may be written 
g-Csec^. Q. if 
G {{sec (ji—sec C} G/Kq if ^ 
The evaluation of the second integral is more difficult; we have 
f"e-““V (X) dX = re-“'“*[2-/(K„X)-/{K.(H-X)}] dX. 
*' u U J 0 
Writing Ki^X = u, the above integral takes the form 
(48) 
(49) 
1 
2K.C 
(1-g-Csec^) 
sec cj> 
fC 
— e~"^'^°^f{u)du — e' 
Jo 
C see <ti I gM sec <l> 
f{u) dll 
We denote by B(x) and B( —a;) the functions 
B {x) = Ei (—x) —log X, B (—a?) = Ei (a;)—log x. 
The expansion when x is small of the exponential integral is 
Ei{x) = y+^logx* + x + 
/Y»3 
*A/ «A/ %Ay 
+ TT—r“. + 
2.2! 3.3! 4.4! 
(50) 
(51) 
(52) 
where y is Euler’s constant, y = 5572 and the expansion holds for both positive and 
negative values of x. 
Thus when x is small the expansions for B(a)) and B(—a;) are 
B (a;) = y— a; + 
x" 
x^ 
2.2! 3.3! 
+ . . • 
and 
/yt2 ^3 
B (—a?) = y + a; + —^ 7 -, + ... 
2.2! 3.3! 
(53) 
It can be shown that the integral 
j* f {ax) dx {l—e~'"^f{ac)} 
Uf 
1 1 
1 1 a 
— loo' - 
h ^ a + b\ h 
— ye ^'^Ei{ — ac)+^Ei{ — {a+h)c] 
(54) 
The result holds for positive as well as negative values of h provided the argument 
of {a + h) c in the logarithm and in Ei {—(a+6 ) c} is the same. 
By means of this result and in the notation defined in (51) it can be shown that 
(50) reduces to 
. 
3 D 2 
(55) 
