6 
Dividing both sides by k and diminishing k without limit, 
the resulting equation will be 
T d(f>(x) _ 
io ~dF "" 
WW 
Eliminating I 0 and I 0 ' by (5) 
the integral of this is 
log(j)(x) = <p\o)% + c 
or 
<f)(x) = Qe^i°) x 
when x — o <f>{x) = 1 
Therefore C = 1 
Also as x increases the intensity diminishes, therefore (p'(o) 
must be some negative constant, let it be denoted by — m. 
Then the equation becomes 
< p(x ) = e~ mx 
and equation (1) may be written 
T _ T P — mx 
Hence experiment leads to the same form for the function 
as the hypothetical form with which we started. 
If in the above investigation we had made the length of 
the column invariable, and x denoted a mass of some colour- 
ing matter which undergoes no decomposition on dilution, 
we might have obtained experimentally the form of the 
function expressing the intensity of the light transmitted 
through a column of fluid of invariable length containing a 
variable quantity of colouring matter. 
In the above remarks I have supposed we are dealing 
with homogeneous light or with white light which has pene- 
trated a medium containing soluble black in solution. To 
apply the formulae generally we must prefix to them the 
sign of summation. 
In seeking a priori the law of transmitted light we might 
have reasoned as follows, which involves less assumption 
