ON THE THICKNESS AND SURFACE TENSION OF LIQUID FILMS. 
635 
If p be the pressure due to the film, T the tension, and cr the density of the liquid 
in the manometer, 
y> = 4T/R=2crp/. 
Hence 
I1_ 2T _ r a( 1+ tt 8 ) 
agl 
( 2 ) 
From these equations we get 
2Tdl/dT-- 
2nr^(l + n 2 ) 2 
°77 
— r„ 4 ( 1 + n 2 f - 2r l ~(n 2 -1) 
This expression gives the sensitiveness, since the total movement (down and up) of 
the liquid in the manometer is 2 dl. 
In Plateau’s experiment T = 60 dynes (about) per linear cm., r 1 =0 - 5 cm., 
r 2 =0’25 cm., cr— 1. 
Substituting these values in the denominator, and equating to zero, we get 
(i + rff — 31-31 5(n* -1) = 0. 
This equation in n 2 has one negative and two positive roots. The latter are 
n 2 = 1-497 and n a =2'912, 
which give 
1 '224 and 72=1-707. 
When n has these values the sensitiveness is infinite. The meaning of this result 
is as follows. If a bubble be blown at the end of a tube connected with a manometer 
the bubble will increase, and the liquid in the manometer will become more depressed 
until the film is a hemisphere. Thereafter the pressure will diminish, and though V 
will increase v will become less. If the radius of the manometer tube is such 
that the decrement dv is greater than the increment dV, an increase in the size of 
the bubble would necessarily be attended with a decrease in the total volume of 
enclosed air. Under these conditions the equilibrium would be unstable, and the 
system would transform itself into a form in which the volume of the bubble was 
such that dV-\-dv was positive. If the bubble were originally on the borders of 
instability a small change in T would produce a finite change in the position of the 
liquid in the manometer, and thus the sensitiveness would be infinite. In the case of 
the apparatus used by Plateau the whole phenomenon must have occurred within 
narrow limits and would easily escape attention. 
The following table explains itself, and illustrates the fact that as the sagitta 
increases Y+ v has successively a maximum and a minimum value. 
4 M 2 
