3 
will destroy some fraction of its whiteness ; let this fraction 
be n. Then the remaining whiteness will be Wo - Wo n, this 
we may denote by Wi, and the mass will be M+m. 
Suppose we repeat the addition of the black, the propor- 
tion being as before M ; m. If x denote the black to be 
added, we shall have the proportion 
M + m;M| \x\m 
Whence 
x — 
(M + m)m 
M 
After this second mixture the whiteness will be Wi - n; 
this we may denote by W 2 ; it may also be written 
Wo(l — ny by substitution for Wi, or still more briefly 
W 0 R 2 when R= (1 — n). 
Let the operation be repeated a third time, the proportion 
of the white mass to the black being still M ; m. After the 
second mixture the mass became it ' So if x denote 
M 
the quantity of black to be added we shall have 
(M + m)\ 
M 
M \\x\m. 
Whence 
x = m + 
2m 2 m 3 
ll + M 2 
The mass will now become 
(M + m) 3 
M 2 
If W 3 denote the whiteness we shall have 
W 3 = W 2 - W 2 n = W 0 R 2 (1 - n) = W 0 R 3 
If we continue the operation n times, then from the 
above law, if Wn denote the remaining whiteness we shall 
have W^=W 0 R n . 
Also the mass will be 
M n_1 
I also used an independent method of reasoning. Suppose 
we have a white area A, then the quantity of white light 
given off in any direction, say normal, to the surface will be 
proportional to A; so that if Wo denote the white light we 
may write Wo=/*A. Suppose now a great number of black 
