634 
PROFESSORS A. W. RE1NOLD AND A. W. RUCKER 
diameters of the films, furnish the required indication. Let an increment dT in the 
surface tension T produce an alteration cZL in this length. The fraction Tc/L dT may 
then be conveniently taken as a measure of the sensitiveness of the experiment. 
The quantities dh and dT may be finite or infinitely small. In the latter case the 
expression gives the limiting sensitiveness for an infinitely small variation in T. 
If the increment dh observed in any experiment is divided by the sensitiveness, the 
quotient is the fraction of itself by which the tension has altered. The relative values 
of different methods of measuring dT is best obtained by comparing the sensitivenesses 
for equal values of d'T/T. 
It is easy to see that, as has been stated, experiments can be arranged in which 
the sensitiveness is infinite, i.e ., in which the equilibrium is unstable, so that an 
infinitely small change in T would produce a finite change in L. Two bubbles, each 
greater than a hemisphere, and the interiors of which were connected, would form 
such an arrangement. Great sensitiveness may be obtained by approaching the 
conditions of instability as nearly as may be practically convenient. 
In the following calculations we have treated the air as incompressible by the small 
forces due to slight changes in the surface tensions of the films by which it is enclosed. 
In Ludtge’s experiment, if the volume of the tube is large compared with that of the 
spherical segments, it might be necessary to allow for variations in the density of the 
air. If the apparatus be properly designed, the necessity for this complication is 
avoided. The surface tension of water is about 81 dynes per linear cm. A water 
bubble 1 cm. in radius would therefore exert upon the enclosed air a pressure of 
324 dynes per sq. cm. The changes of surface tension to be studied are but small 
fractions of the whole, and as the atmospheric pressure is 10 6 dynes per sq. cm. it 
may safely be said that they could not alter the volume of the enclosed air by more 
than a few millionths of the total space occupied. 
In experiments in which the equilibrium is stable the change of form produced by 
the change in surface tension is generally such as to reduce the change of pressure 
which would otherwise be produced. The contraction or expansion of the enclosed 
air would also affect both films, and its effects would be the same as those of a small 
change in temperature. 
Limiting Sensitiveness of Plateau’s Experiment. 
Let Y be the volume of the bubble, r x the radius of the manometer tube, and l 
the distance through which the liquid is depressed. Let r 2 be the radius of the tube 
from which the bubble hangs, It the radius, and nr 2 the sagitta of the bubble. 
Then 
R= r 2 ( 1 + w 2 )/2 n, Y = 7rr 2 s n(n z -f- 3)/6. 
Also, if v=Trr 1 2 l, we must have 
V+ v— a constant 
(1) 
