ON THE THICKNESS AND SURFACE TENSION OF LIQUID FILMS. 
641 
same as the curvature of the circle the diameter of which is the major axis of the 
generating ellipse in the one case, and the transverse axis of the hyperbola in the 
other. 
Now, since a and f3 are the maximum and minimum ordinates, it is evident that in 
the ordinary notation 
a — a( 1 + e), /3=ct( 1—e). 
Hence 
1 + 1 =- : 
R t R / a 
« + /3 
If then a and /3 alter, the increment in the curvature is 
— 2 (da +cZ£)/(a-f ft) 2 . 
If the variations take place subject to the conditions that X and Y are constant, and 
if (8) be written in the form 
Mc/a-fi a 3 Nc//3=0, 
we have 
(« + £)V l 
“2“^E 
‘k 
« 2 N-M , a 3 N—M ln 
aot= ——— dp, 
« 3 N 
M 
(ii) 
In the case of a spherical film 
(3= 0 , Jc 2 = 1 , A cos (fi v E= sin <£ ]5 F=log e tan (J + 'y 
Hence 
M= — a 3 /sin (f) j, N= — log, tan + p), 
and from (11) 
dB + R')-i 
cosec — log, tan j 
log, tan (j+j 
The denominator of this expression is positive, and possible for positive values of 
<^< 77 / 2 . The numerator may be shown by trial to be positive for values of 
4>i<5 6° 28'. When this value is reached it becomes zero and changes sign. 
Now in the case of a sphere 
X=asin (f> 1 ; and, since a 3 =X 3 +Y 3 , Y=Xcot<£ 1 . 
Therefore, if <^ = 56° 28', 
and 
2X=3-0179Y=0-4803X2ttY, 
a=l-8102Y. 
4 N 
MDCCCLXXXVI. 
