o 
But suppose we had started with no hypothesis as to 
the form of the function expressing the intensity of trans- 
mitted light, hut had found as an experimental result that 
when the intensities are equal they remain equal when the 
columns receive equal increments ; what form for the func- 
tion might he deduced from such a result ? 
Suppose there are two lights of initial brightness I 0 and 
I 0 ' respectively, then if x and y he the lengths of the absorb- 
ing columns, for the transmitted light in one case we shall 
have 
l x = l 0 <p(x) (1) 
and in the other case 
i y = (2) 
being the unknown function which it is required to de- 
termine. Since these two intensities are equal, 
I 0 (f>(oc) = l 0 '<p(y) (3) 
If both columns receive an equal increment, the intensities 
will again be equal. Let this common increment be de- 
noted by k, then : 
+ k) = iMv + k ) (i) 
These equations will still hold if y—o, In this case the 
duller surface is observed directly, and the length of the 
absorbing column over the brighter surface is increased 
until they are brought to the same intensity. Since when 
y—o therefore ^(o) = l. 
Equations (3) and (4) will become 
I 0 <p(x) = 1/ (5) 
I 0 (p(x + ic) = I </0(k) (6) 
Expand (6) in terms of k 
a**) + ^ + w* 2 + &o - I;(1 + ^ + + *<=•) 
Since the first terms of the expansions on each side are 
equal, the equation may be written 
U 
fdfyx d 2 (f>x 
V dx K + 2 dx^ 
+ q^ +&c , 
0 
