80 DR. JAMES BOTTOMLEY ON THE 



mix with grey, will not be the same as if we mixed with black. 

 The formula will have to be altered as follows : — Suppose 

 W the initial whiteness, after the addition of perfect black 

 the residual whiteness will be W R n ; but if the material 

 added be grey, we must give back a quantity of white, 

 which will be some fraction of the whiteness lost. Suppose 

 p to be this fraction ; also the whiteness lost will be 

 W — W R W , so that the quantity of white to be restored 

 will be W p(i — R tt ), and the total whiteness will be 

 W R n (i — p) +W Q p. If we suppose n to become infinite, 

 the whiteness becomes W p, being the whiteness or grey- 

 ness of the so-called black body. We might also have 

 deduced the formula as follows : — Take a white area A, 

 then the quantity of white light given off we may denote 

 by fiA. Now let a series of grey points fall upon this ; 

 let a be the area of the spots ; then the quantity of white 

 light given off by this we may denote by //.,#, the uncovered 

 white area will be A — «, and the quantity of white light 

 given off by this will be ft (A— a); therefore the whole 

 quantity of white light will be fi(A—a) + /*!«, or fiAR + fjb t a 



if R be written for i — -r-. If we suppose another series of 



grey points distributed over the surface, the uncovered 

 white area will be AR 2 , and the surface covered by the 

 grey points will be A— AR 2 ; so that the quantity of light 

 will be //A.R 2 + /jl 1 (A — AR 2 ) . If the operation be repeated 

 n times, the expression for the residual whiteness will be 

 //AR^ + z/^A— AR n ), which maybe written in the form 



fiA[U n {^ — p) +p\, where;? = — ; also fiA=W , the initial 



whiteness ; so that the expression is equivalent to the one 

 previously given. 



