OjST the thickness and surface tension of LIQUID FILMS. 665 
Now, if we are dealing’ with one film only which is not connected with another, and 
if it would assume the cylindrical form when of uniform surface tension, dV—0. In 
this case we get from equations (26) and (27) 
tan 2(£ 3 sin 2£ 3 —1 + cos 2£ 3 tan 2</> 2 cos 2£ 3 — sin 2£ 2 tan2<£ 0 —2|f a 
—tan 2 $! sin 2 ^— 1 + cos 2 ^ tan 20 x cos 2^+sin 2^ tan 2<£ 1 + 2 £ 1 ’ 
and, solving these equations, we get 
tan 2(f)) = 
tan 2 <f> 2 = 
fi sin 2 & - sin- fi sin (& + 2| 3 ) + sin & sin g, sin (& + g 3 ) 
sin (& +1 a ) { £ a cos (& + £ a ) - sin f s cos & } [ 
f 2 sin 2 — f, sin sin (g, + 2+ sin f L sin g 3 sin + £,) 
sin + cos (?i + ^)-shi & cos £ a } 
(28) 
If dp is the difference between the internal pressure and that exerted by a 
cylindrical film of radius Y, 
or, from (22), 
dp— -^(Ja 1 + c//3 1 ) = 2 | T -^ 3 (^aofi 
7 T cos 2 6 -, , 2dT T cos 26., 
dp—— — j do. i=—+— 
1 Y 2 sm- <f>i i Y 2 sin 2 </> 2 
ff/3,), 
da 2 , 
The angles </> 1 and (f> 2 can now be found from (28). By means of (25) and (29) 
da.) and da.., are given in terms of dT, and thus d/3 y and d/3% are known from (22). 
We shall suppose that the suffix 2 refers to the black part of the film. The general 
nature of the changes of form which the film would undergo as increased can 
easily be deduced from the equations. When £ 2 =0, tan. 2<£ a =0, tan 2^ = 0/0. 
From the first of these conditions we deduce <^ 3 =0 or ir/2. The first of these 
values corresponds to the case where dT is negative, when the black part of the 
surface would bulge ; the second to the case where dT is positive. Since in each case 
the tangent to the unduloid is parallel to the axis, an infinitely thin ring, of different 
surface tension to the rest of the film, would not cause any finite change of form. 
Evaluating tan 2</> l5 we get 
tan 2^= (sin 2^—2^)/2 sin 3 
which gives the initial value of (f> v 
The distance from the point halfway between the rings of the point at which the 
initial unduloid would cut the generating line of the cylinder is Y(2^> 1 — ^), where 
<f>i and have, of course, their initial values. 
If £) and £/ be such that and £, = £/, it is evident from (28) that 
tan 2<£ 1 =—tan 2<ft. 2 ' and tan 2^> 3 =— tan 2<£/, so that </»i-h 7r /2. 
MDCCCLXXXVI. 4 Q 
