ON THE THICKNESS AND SURFACE TENSION OF LIQUID FILMS. 
663 
respectively, then X x and X 3 may be considered as constants which, are given 
by the conditions of the problem. 
(2.) The values of the ordinates which correspond to P and Q must both be Y. 
(3.) The ordinate of R(=y) must be the same for both curves. 
(4.) The tangent at It must be the same for both curves. 
(5.) The sum of the volumes of the two unduloids contained between the rings may 
be taken as given. 
(6.) The pressures exerted by the two parts of the film on the enclosed air must be 
the same. 
These eight conditions may be used to determine the eight unknowns, viz., da x , d/3 x , 
da. 2 , d/3. 2 , <f>i and </> 3 , the values of <f> which correspond to P and Q, and i/q and i// 3 , the 
values of xjj which correspond to P, considered as belonging in the first case to PR, 
and in the second to PvQ. 
The discontinuity at R is dependent on the fact that in the cylinder the value of <f> 
which corresponds to any given point is indeterminate. We shall carry out the 
calculation to the first order of small quantities only. 
Using the same notation as before, we have, as in equations (15) and (16), 
7 2 _ 2(dct 1 — d/3 l ) , 3 _2 (dct 2 —d/3 z ) 
' L 1 y ’ — y s 
(19) 
<£o — r// 3 =£j 1 
2Y J 
clc^ + dfr jl 
2Y j 
(• 20 ) 
In like manner we get from (5) 
Y 1 /7r=2a 1 3 (l-|^ 1 2 )(T/; 1 -<^ 1 )+^Y L (sin 2i/q— sin 2^), 
Li 
whence, substituting from (19) and (20), 
Y 1 /^=2Y^ 1 (n-H±^^ + Y^ 1 -rf/3 1 )(siu 2^,-sin 2*). 
But 2vY 3 £ i — TrY z X |, so that, if dY be the difference between the sum of the volumes 
of the figures generated by PR and RQ and the volume of the cylinder, 
dY/7rY 2 = 2£ 1 (du 1 -f d/3 x ) -j- (da l — c//3 1 )(sin 2xfj 1 — sin 2</> 1 ) 
+ 2^ 3 (coa 3 -f- d/3%) -f (da 2 —d/3 3 )(sin 2 <f>. z sin 2 1 // 3 ).(21) 
