116 



is such, that after running through the element in question the 

 diffusion of the rajs lias only taken place over a small cone. A 

 characteristic difference with molecular scattering is, that with the 

 scattering by refraction not a great part of the bundle goes on 

 unimpeded and a small pai-t spreads to all sides, but that the chief 

 bundle itself gets continually broader. 



And so, expressed in mathematical terms: let us follow light of 

 a given direction over a length /, then, if this bundle has an 

 intensity of one per unit of square, the intensity of the light which 

 is found in a cone of the opening r/tu, of which the axis formed an 

 angle « with the direction of incidence, will be possible to be 

 represented after running through / by a function : 



/ («,/) dio or X W) d^ 



By making use of a particular image the form of the function -/ 

 may be determined. This form will be analogous to the law of errors; 

 it is however unimportant for what follows. What is important is the 

 supposition that the function •/ possesses a perceptible value only 

 for very small values of a. 



If we take the meaning of •/ into consideration, we see imme- 

 diately that of course 



ƒ 



X («) doi = 1 



where the integral, just as evei-y where else in what follows, must be 

 taken over the whole unity S[)here. 



Now we shall deduce the integral equations for the intensity of 

 radiation. Let / (r, ?/, 2, »>, 7') represent the intensity of radiation in a 

 point {x,y,z), whilst the direction is given by the angle i) with the 

 ,r-axis and (f . If now we know the radiation in a point (.r, ?/, z), 

 we ask this quantity in a point that is situated / further in the 

 direction of the ray i'>, if'- The coordinates of this point ai'e : 

 ,); '^^ I cos d- , y -\- I sin d- cos <p , z -\- I sin i> sin (f 



And so the intensity of radiation may bo represented by : 

 f {x -{ I cos 1^ , y -\- I sin x> cos (f , z -]- I sin 0- sifi (f , i> , q). 



This intensity must now be equal to the intensity which by the 

 bending of rays comes in the given direction. When iy' and \\>' are 

 the angles which detei-mine a ray in x,y,z, then, if a repi-esents 

 the angle of this ray with the ray {>, \]\ the intensity in the second 

 point will also be given by : 



/ 



'/^{ft)f{.r,y,z,i)-\q')dtiJ 



