74 DR. JAMES BOTTOMLEY ON A 



such bias, and that they will as readily fall upon a black 

 as upon a white surface. Now the surface on which they 

 fall is grey or a mixture of black and white. So we have 

 this question in the distribution of the second black area 

 (consisting of innumerable detached points) : — How much 

 falls on the black surface, and how much on the unoccupied 

 white area ? Let p be the portion that falls on the black 

 surface, and q the portion that falls on the white ; now 

 what will be the ratio of p to q ? If we suppose the second 

 series of black points to be fairly distributed, the portion 

 which falls on the black surface will be to the portion which 

 falls on the white as the areas of those surfaces, so that 



p _ a 



(0 



q A — a' 

 also 



P + q = a, (2) 



and the remaining white area will be A — a — q. From ( i) 

 and (2) we have 



qa 



A— a 



a-q } 



whence 



a(A-fl) 



A ' 

 and for the remaining white area we have the expression 



A-a- a ^- a) =A(r-ff=AK\ 



and, the quantity of white light given off will be /xAR z ; 

 also this quantity of light will appear equally distributed 

 over the whole surface ; hence, if W 2 denote this white 

 light, we may write 



W 2 ,= f iJiR z =W Q R\ 



