ON THE THICKNESS AND SURFACE TENSION OF LIQUID FILMS. 
643 
and from (8) 
so that 
da d/3 da + d/3 
tan <^> 1 — cot cpy —2 cot 2 
4daT/(/T=—4Y sin 3 0 x /cos 2 (f> l . 
This expression gives the limiting sensitiveness, since the total alteration in the 
lengths of the two diameters is 2(cZa— d/3') = Ada. 
If the films were initially spheres and the arrangements were in all respects the 
same for both we have den— —da, d/3=—d/3'. Also, substituting the proper values 
of E and F, given above in equation (8), 
da 
d/3 
da + d[3 
log, tan 7 + 
9r , 0i 
cosec 0! , i 17 \ 0i\ 
log, tan I —) — cosec fa 
whence, and from (12), if we put <iT' = 0, 
4daT/dT= 
• 7T ( 1 ) i 
la log, tan (.Ay 
/ 7T . (^) j » 
log* tan ( - “t——) — cosec </> x 
By the aid of these expressions we have calculated the following table. The radius 
of the rings is taken as the unit of length. The value of X is therefore numerically 
equal to the distance between the rings in terms of their diameters. The values of 
<f>i have been so chosen as to give approximately equal values of X in the parts of the 
table which refer to the cylinders and spheres respectively. 
Table III. 
Two cylinders (Y=l). 
Two spheres (Y = l). 
<px expressed in 
X = 2^. 
idaT/dT. 
<p 1 expressed in 
degrees. 
X = tan (pi. 
a= sec <Pi. 
4daT/dT. 
Circular 
measure. 
Degrees. 
1/8 
746 
0-25 
0-064 
O 
14 
0 
0-249 
1-0306 
0-13 
1/4 
14-32 
0-50 
0-279 
27 
0 
0-509 
11223 
0-64 
3/8 
21-49 
0-75 
0-734 
37 
0 
0-754 
1-2521 
1-80 
1/2 
28-65 
1-00 
1-700 
45 
0 
1-000 
1-4142 
4-68 
5/8 
35-81 
1-25 
4-343 
51 
21 
1-250 
1-6011 
14-24 
7t/4 
45-00 
7r /2 
00 
56 
28 
0-961 x ?r/2 
1-8102 
00 
4 N 2 
