390 
MR. LOUIS VESSOT KING ON THE SCATTERING AND 
remains finite, although its application to the case of the earth’s atmosphere is some¬ 
what invalidated by the curvature of the earth. 
On examining (59) it will be noticed that as long as C is not too large we may 
write 
T (^, f) = ^ sec ^.T (0, f) = ^.T (0, 0, . ■ ■ • (66) 
i-tcos (, 
which may be taken as a rough approximation for the intensity of sky-radiation from 
any direction not too close to the horizon where (65) must be used. When C is large 
the wave-length corresponding is small, and the intensity in the normal solar spectrum 
outside the atmosphere also becomes small. Hence the formula ( 66 ) is sufficiently 
accurate when the total intensity on a horizontal plane is required. Denoting by 
H (i) the intensity of scattered radiation of wave-length X received per unit time on 
unit area of horizontal surface, we have 
H (^'') = [ T ( 0 , 0 cos 0 da), 
V 
the integral being taken over the entme sky. 
Since dw = sin <p d-yp-, we have 
r2ir r'/c"’ 
H (f) = d'P \ T cos<psui(pd<p, .(67) 
Jo Jo 
and making use of the approximation in ( 66 ) we have 
H (0 = r di. r^'"(l +cos^ 0 ) sin 0 d 0 . 
1 “t cos f J0 Jo 
The total intensity for all wave-lengths per unit area of a horizontal surface is 
given by 
27r 
f' H (i) d\ = 
Jo 
f (l-tcos^ ^) 
1*00 
T(0,0 
Jo 
dX. 
( 68 ) 
Of some interest is the intensity of the radiation which is scattered from the 
atmosphere back into interplanetary space. If we write (H —X) for X in (47) we 
notice that 
K ( 0 , 0 = sec 0 r e-i^oxseo^ j (H-X) dX ; 
Jo 
we also notice from (36) that 0 (H —X) = 0 (X). 
We thus obtain from (45) 
(«/*, f) = Mo (^) S sec 0 
which reduces to 
rH 
0 
-KflX (sec(/)-!-sec ^)dX+ 6 
u \ rH 
, —KnX see </> 
0 (X) dX 
R {<),, 0 = 
C sec 0G {C (sec ^ + sec ^)} +§■ M ) *1* (C, ^•) 
(09) 
