J17 

 so that the required iiiteg-ral eiiuatioii is: 



f(x -f I cos d;.,,)= I X («) ƒ (a-, y, z, »', </)') dio'. 



Now it is easy to transfonii this integral eqnalion into a differential 

 equation, if we bear in mind that / has only perceptible values for 

 small values of «. We express by (t the integration-element dta' 

 and the angle if', which the plane thiongh ('^', f/-') and (0,ff) makes 

 with that through (''/(f) and tlie .I'-axis. So that the value of t/to' is : 



sill a t/n dx^y = u da (/if> 



For the difference of the angles and 0' and that of the angle 

 <p and cp' we find going to the second order to « and after elementary 

 reduction : 



. cot^ain* \ï> 

 Lxd-= iy — 0^= — K COS If' + 2 s?;i v> COS xh L\f/> = — a COS \p ~\- a' 



sin \p 

 t\(p = a -7 — . 

 sin ^ 



Now we can develop in the integral /(.r, //, 2, »>', 7/) with respect to 



L^ and Ar/) and get in this way : 



ƒ (a-, ?/, z, », cf) 1 X («) r/to'H- 

 j I A Ö- «X («) dad\l^-\-j- i I Axp «x (^^) da dtp -f 



1 öv' rr , dy 



2 d;^' 



"^ 0^. 



+ - ^ Jj" ^"^'^^^ "'^ ^"^ ^"^ "^"^ + dd^ [f^^ ^'^ ''^ ^"^ '^'' "^"^ 



The first integral is equal to unity, the second yields 



jt r 



-cot^ju'x 



the third is zero as well as the fifth, whilst the fourth and the 

 seventh yield : 



jr 1 «' / («) da and ^— — I a* x (''<') ^^<' 



Now we can introduce the mean value of «* according to 



'c7- z= j «' ^/i{u) do) =2jtI u' X (f^) da 



And thus we obtain at length, — if we also combine the first 

 term of the second member with the first member and develop 

 according to / — for the differential equation of the diffusion of 

 light by irregular refraction : 



