384 On Transmission of Light through an Atmosphere. 



Let us suppose then that a large number of small disks are 

 distributed at random over a plane parallel to a wave-front, 

 and let us consider their effect upon the direct light at a 

 great distance behind. The plane of the disks may be divided 

 into a system of Fresnel's zones, each of which will by 

 hypothesis include a large number of disks. If a be the area 

 of each disk, and v the number distributed per unit of area 

 of the plane, the efficiency of each zone is diminished in the 

 ratio 1: 1 — va, and, so far as the direct wave is concerned, 

 this is the only effect. The amplitude of the direct wave is 

 accordingly reduced in the ratio 1 : 1— va, or, if we denote the 

 relative opaque area by m, in the ratio 1 : 1 — m*. A second 

 operation of the same kind will reduce the amplitude to 

 (1 — m) 2 , and so on. After x passages the amplitude is 

 (1 — m) x , which if m be very small may be equated to e~ mx . 

 Here mx denotes the whole opaque area passed, reckoned 

 per unit area of wave-front ; and it would seem that the result 

 is applicable to any sufficiently sparse random distribution of 

 obstacles. 



It may be of interest to give a numerical example. If the 

 unit of length be the centimetre and x the distance travelled, 

 m will denote the projected area of the drops situated in one 

 cubic centimetre. Suppose now that a is the radius of a 

 drop, and n the number of drops per cubic centimetre, then 

 m = mra 2 . The distance required to reduce the amplitude in 

 the ratio e : 1 is given by 



x = 1/mra 2 . 



Suppose that a=^ centim., then the above-named reduction 

 will occur in a distance of one kilometre (x = 90 b ) when n is 

 about 10 -3 , i. e. when there is about one drop of one milli- 

 metre diameter per litre. 



It should be noticed that according to this theory a distant 

 point of light seen through a shower of rain ultimately become*! 

 invisible, not by failure of definition, but by loss of intensity 

 either absolutely or relatively to the scattered light. 



* The intemity of the direct wave is 1 — 2?n, and that of the scattered 

 light m, making altogether 1—m. 



