84 Dr. W. E. Sumpner on the Diffusion of Light. 



where A is the total area of the bounding surface, of which a 

 portion A x has a reflective power equal to rj^ and a second 

 portion A 2 a reflective power 7? 2 , &c. This relation is very 

 approximately true for ordinary rooms, and may be shown to 

 be quite accurate for a spherical enclosure. 



Fiff. 1. 



For let P and Q be any two points of a sphere of centre C 

 and radius r (fig. 1 ) . Then PQ = 2r cos <£, where </> is the angle 

 which the chord makes with the radius through either P or 

 Q. Also, with the same notation as before, we have 



t / t , f d ,dA cos 6 



PQ' 2 



where dA is an element of area at Q of brightness B, 

 and subtending a solid angle <iAcos<£/PQ 2 at P. Now 

 PQ 2 = 4?' 2 cos 2 <£ and 7rB = 7]V, where I' is the illumination of 

 the area dA : hence 



V= I P +lij 1 VA = I p+ ijlVA, 



(6) 



A being the total area of the spherical surface. The integral 

 is constant whatever the position of the point P, and what- 

 ever the character of the reflecting surface of the sphere. 

 Thus if any complete [or if any portion of a] spherical sur- 

 face be illuminated in any manner J p by the direct rays of a 

 combination of light sources, the actual illumination 1/ will 

 exceed J p by a constant amount all over the sphere, owing to 

 the reflective action of the surface. Also, if the original dis- 

 tribution be uniform all over the sphere, I p = I, a constant, 

 will also be constant, =P, and 

 j/ 

 + ^jVA= 1 + Vm V, 



V 



I' = I 



where 



V m = 



ZydA 



(?) 



