ON THE THICKNESS AND SURFACE TENSION OF LIQUID FILMS. 
053 
Hence, if we neglect the squares of small quantities, 
^=(^-/3 2 )/od=2(da-d/3)/Y .(15) 
If x be the difference of the abscissae of two points for which <f> 1 and <jY are the 
values of <f), then, to the same approximation, 
Let 
Then 
= 4' A d<f,+4l f=2«(l -f )(*i •- <k)= 2Y {14 
*/ 2 Y=f. 
da + d/3 1 
2Y 
• ( 1G ) 
Now from equation (4) 
da + d/3 ] 
2Y r 
(Y+c/y 1 )~ = (Y-[-c?a ) 3 cos 3 ^> 1 +(Y^+c?y 8) 2 sin 2 <^ 1 , 
or, neglecting the squares of small quantities, 
7 da + d/3 da — d/3 . 
c «/i= +— 5 —cos 2<j) v 
(17) 
If now we take three points on the unduloid, such that the distances between con¬ 
secutive points measured parallel to the axis are equal, then to the first approximation 
^i — <f>-2=<l>-2—<k=& 
But, from two equations similar to (17), 
Also 
dy l + dy%=da-\- d(3 + [da — d/3) cos 2 <f>. 2 cos 2 £ 
7 da + d/3 da—d/3 n . 
^ 2 / 2 =—K— H- 9 cos 2 (f>,. 
Multiplying the last equation by 2 cos 2 £ and subtracting, 
cict-\~ df3- 
dy l + dy s — 2dy 2 cos 2f 
2 sirdf 
Since da and d/3 are independent, we may suppose the unduloid to have been 
derived from any convenient cylinder, and the calculations are simplified if we select 
that the radius of which (Y) is the mean of the three quantities Y-f -dy x , &c. 
