72 DR. JAMES BOTTOMLEY ON A 



white it will destroy some fraction of its whiteness ; let 

 this fraction be n. Then the remaining whiteness will be 

 W — W Q n ; this we may denote by W, ; and the mass will 

 be M + m. 



Suppose we repeat the addition of the black, the pro- 

 portion being as before, M : m. If x denote the black to 

 be added, we shall have the proportion 



M + m : M : : x : m ; 

 whence 



(M + m)m 



x= 



M 



After this second mixture the whiteness will be 

 W 1 —W l n; this we may denote by W 2 ; it may also be 

 written W (i—n) <2 by substitution for W l} or still more 

 briefly W K 2 when R= (i -rc) . 



Let the operation be repeated a third time, the pro- 

 portion of the white mass to the black being still M : m. 



After the second mixture the mass became M — — . So, 



if x denote the quantity of black to be added, we shall have 



Whence 



M 



M 



: : x : 



m. 



a?=m + 



2m z 



w 



m 3 



i 



The mass will now become 



(M + m) 



M 2 

 If W ? denote the whiteness, we shall have 



W i =W 1 -W z n=W R z (i-n)=W RK 



