THICKNESS AND ELECTRICAL RESISTANCE OF THIN LIQUID FILMS. 525 
be sharply distingalshed from each other, and whose properties are different, it is 
easy to express the relation between the optical and electrical thicknesses by a 
formula. 
Let T and x be the total thickness and the thickness of a surface layer of the film 
respectively. 
Let K, li, and k be the conductivities of the liquid in bulk, of the interior of the 
film, and of the surface layer respectively. 
Then if T, be the thickness deduced from a measurement of the resistance we 
have, if 
T > 2 a:, 
, Te _{T-2x)k+2XK 
’ “ T “ TK 
Hence 
rp TZ: 2x {k — h) 
' ~ ~K K 
If, now, the results of the experiments be plotted with T, and T for ordinate and 
abscissa respectively, the formula shows that— 
(1.) If the film be of constant conductivity equal to that of the liquid in bulk, 
the results will be represented by a straight line through the origin inclined 
at 45° to the axes. This follows from the assumption K = ^ = /c. 
(2.) If the conductivity be constant throughout the film, but different from that 
of the liquid in bulk, the line will pass through the origin, but will not be 
inclined at 45° to the axes. 
In this case K ^ ^ and h = k. 
(3.) If the conductivity of the surface layers be different from that of the 
interior Jc, and the line will not pass through the origin. 
(4.) If the conductivity of the interior be variable, and if k can be expressed by 
a formula of the type 
^ = A’o + + &c.. 
the expression becomes 
T = 
K 
I 
All terms similar to that in which A occurs vanish when T = 2x. Hence if n be 
large, the term in T" will be small when T is large, and will vanish when T = 2x. 
Thus a straight line will pass through the points corresponding to large values of T 
and through the point corresponding to T = 2 x. For intermediate values the points 
given by the observations will not lie on the line in question. 
