426 Mr. J. C. McConnel on the 



alternating with vanishing-points. But for the present let us 

 make abstraction of these variations. Let 2y = -^r, so that 7 

 is the complement of the angle of incidence of the light from 

 the centre of the solar disc, and let q be the area of the plate, 

 A the intensity of sun-light, and co the angular area of the 

 sun. The light that falls on the plate is q sin 7. A. This 

 light is distributed over an area r 2 co near the observer, at the 

 distance r from the plate. Hence the intensity due to one 

 plate at that distance is qsmy .A/r^w. If there are n suit- 

 able plates within the area co, the brightness of the cloud is, 

 in terms of that of the sun, nq sin y/r^co. We proceed to 

 investigate the value of n. 



With the aid of a little spherical trigonometry, we may 

 show that every plate will reflect light to the eye, whose 

 normal lies within an angular space of approximate area 

 &)/4 sin 7. Now w T e assume for simplicity that the plates are 

 arranged at random, so out of the whole number a fraction, 

 6)/87T sin 7, will send light to the eye. For it must be re- 

 membered that each face is competent to reflect. The average 

 apparent area of a plate is q/2 1 so, if the plates occupy a 

 fraction « of the field of view, the whole number in the area 

 co is 2cor q a/q, and the number in position to reflect light to 

 the eye is n = co' 2 r 2 a/4:7rq sin 7. Hence the brightness of the 

 cloud is o)a/47T that of the sun. Inserting numerical values 

 we have 



•0000045* (1) 



We must now take into account the varying intensity of 

 the reflected light according to the angle of incidence. It is 

 only with the maxima that we are concerned. At each 

 maximum we have for this intensity 



AV 2 



I= (l + 6 2 ) 2 ' (2) 



where V 1 is the proportion of light reflected from a single 

 surface, and the intensity of the incident light is unity. By 

 Fresnel's laws, which are sufficiently accurate for our present 

 purpose, 



sin 2 ((£-(/>') tan 2 (</>-<£') 



sin 2 ((/) + f) tan 2 "((/) + ^)' ' ' * ■ W 

 From these formulas I have calculated the following table : — 



Table I. 



fhen-f= 0° 



<£ =90° 6 2 =1 



and 1=1 



„ t=20° 



«£ = 80° 5 S = -327 



» 1= -76 



„ t=30° 



«/> = 75° 1?= -20 



„ 1= -56 



>, ^ = 50° 



= 65° 5 2 = -08 



„ 1= -275 





