ON THE THICKNESS AND SURFACE TENSION OF LIQUID FILMS. 
637 
If one of the films be maintained in a constant state the increment of T for that 
film is zero. 
Table II. (n= 1.) 
n 
■2'Mn/dT 
0 
0 
0-25 
0-2833 
0-50 
08333 
075 
2-6786 
1-00 
CO 
If n > 1 the arrangement is unstable. 
Van deb. Mensbbugghe’s experiments described above are. from the mathematical 
point of view, particular cases of Ludtge’s method, and do not require separate 
discussion. 
On the Limiting Sensitiveness of the Experiments described in this Paper. 
In most of the experiments hereafter described we have balanced two cylinders 
against each other. Such figures deform into unduloicls, the equations to which could 
be expressed by the approximate relation 
y=ci-\-c sin 
x 
d 
We have also balanced spheres against spheres, and a sphere against a cylinder, 
and it therefore seemed better to deduce general expressions from which the 
sensitiveness could in each case be calculated. 
Beeb # has shown that, if the axis of x be the axis of revolution, the equation to the 
curve which generates a surface of revolution of constant curvature is 
dx— i 
dy. 
( 3 ) 
In this expression a and /3 are both taken as positive quantities, and the curve is 
an unduloid or nodoid according as the positive or negative sign is taken before the 
product a/3. We shall omit the alternative sign and, taking a positive and always 
numerically > /3, shall assume that /3 may be either positive or negative. We shall 
also assume that normally the origin lies on a maximum ordinate. In that case dx/dy 
is negative for positive values of y which correspond to values of x less than that for 
which the first minimum value of y occurs if ft is positive, and less than that for which 
y 2 d-a/3=0 if /3 is negative. In most of the cases with which we have had to deal 
* ‘ Tractatus cle theoria mathematica phaenomenorum in liauidis actioni gravitatis detractis observa- 
torum,’ Bonn, 1857. See also Plateau, ‘ Statique des Liquides,’ vol. 1, p. 139. 
