428 Colours diffusely reflected from some Collodion Films. 



When the obstacle is a sphere, the integral in (3) can be 

 evaluated*. The centre of the sphere, of radius R, is taken 

 as the origin of coordinates. It is evident that, so far as 

 the secondary ray is concerned, P depends only on the 

 angle (x) which this ray makes with the primary ray. We 

 suppose that % = in the direction backwards along the 

 primary ray, and that % = 7r along the primary ray continued. 

 Then with introduction of the value of h from (1), we find 



p _ _ AK . 47rR 3 . **(»*-*»•> /sin in cos m 



mr 



), ■ (5) 



K V m* 

 where ro = 2nR cos \ X (6) 



The secondary disturbance vanishes with P, viz. when 

 tanm = m, and on these lines the formation of colour may 

 be understood. Some further particulars are given in the 

 paper just referred to. 



The solution here expressed may be applied to illustrate 

 the scattering of light by a series of equal spheres distri- 

 buted at random over a plane perpendicular to the direction 

 of primary propagation. The effect of a reflector will be 

 represented by taking, instead of (1), 



h = e int (e inx + e- ik ( x ~ x ^) } (7) 



# expressing the distance between the plane of the re- 

 flector and that containing the centres of the spheres. 

 The only difference is that m~ B sin m — m~ 2 cos m is now 

 replaced by 



sin in cos in 

 . _j_ p i 



„ /sin m' cos m'\ /0 . 



where in is as before, and m' = 2nR sin ^y In the special 

 case where, while the incidence is perpendicular, the scattered 

 light is nearly grazing, x = ^7r, sin |%= cos ^%= 1/^/2, and 

 m = m' = s /2 . IcR; so that (8) becomes 



(l + /*,)/2JE£_£2^ (9) 



\ m d m- j 



This vanishes if cos kx Q = — 1 ; otherwise the reflector merely 

 introduces a constant factor, not affecting the character of 

 the scattering. At other angles the reflector causes more 

 complication on account of the different values of m and m! . 



October 2, 1917. 



* Proc. Hoy. Soc. A. vol. xc. p. 219 (1914). 



