16 
In the discussion which followed it was suggested by Mr. 
Heelis that some of the spots might be superimposed on 
others in the same unit. This will leave the expression for 
W n unaltered in form, but will introduce a new value for 
the constant. The symbol It stands for 1 — A being the 
area of the white surface and a the surface occupied by the 
mass of black. The problem under consideration is a physi- 
cal one, and ultimately the spots will be due to the atoms 
of matter ; these are finite in magnitude. Let p be the area 
covered by an atom of matter, and suppose our unit of mass 
to contain p atoms — let these be thrown down singly on 
the surface ; then the remaining whiteness after the expen- 
diture of the unit will be Wo^l - If we throw 
down n units the remaining whiteness will be 
Wo 
(>-i) 
This then will be a strict solution of the 
problem, for manifestly one particle cannot be superimposed 
on itself. If we keep to the same unit of mass and the 
same kind of matter, we may write It for ^1 - jQ so that 
the expression for the residual whiteness may be written 
WoR.% being the same in form as that given before. 
We may also write our first expression in the form 
W^=Wo £ v a) 
or if we expand the logarithm 
( —a a? \n 
A ZE? &c 7 
on a particular hypothesis a simple expression may be 
obtained for the residual whiteness due to the distribution 
of a given quantity of matter over a surface. Suppose j 
that a can vanish and n become indefinitely great, the 
first term in the index is — -Jf. Hence this multiplied by n 
would tend towards the ambiguous form ° ^ ; but the limit 
