640 
PROFESSORS A. W. REINOLD AND A. W. RUCKER 
Whence, substituting as above for the integrals, we get 
dV/TT^da 
-j-c//3 
u—/3 
{(2a ? —/3 : )E—F/3 3 } +a 3 a 1 sin fa cos fa 
^-^{(a 3 —2/3 3 )E-f-F/3 3 } — a/? a x sin fa cos (/>j 
A 1 a' 2 (a+/3)( 1 — -—— sm~fa 
If Y is constant this becomes from (7) 
dV/n — — —g[cZa{(2a 3 — /3 3 )E — F/3 3 fi- A 1 a 3 cot (/q) 
+ ^ {(or — 2/3 2 )E-f-F,d 3 + a i/3 3 tan </q}],.(9) 
and, if both X and Y are constant, we get from (8) 
dY/n = daX ^-—^{(2Ea 3 —F/3 3 + A Y od cot </q)(2E — F— A x tan fa) 
— E 3 c4 3 } -p(E — F— Ai tan fa) 
= —dfiX a(a+y8){(2Ea 3 —F/3 3 + A x a 3 cot ^> 1 )(2E — F— A 1 tan 
—E 3 a 3 } -i- (F— Ect 3 — a 3 A 1 cot fa) .(10) 
These equations applied to the problem under discussion enable us in the first place 
to determine what arrangements of two films in communication will be stable. 
Maxwell # has shown that a slight bulge in a cylinder will increase or decrease 
the pressure according as the cylinder is shorter or longer than half the circumference 
of the rings to which it is attached. Hence two cylinders of equal length and radius 
will be unstable if their lengths exceed the limit thus laid down. For, if one cylinder 
expands and the other contracts a little, the pressure in the first will fall off, and that 
in the second will increase. Hence, the tendency will be to depart still further from 
the position of equilibrium. We found (as w r as to be expected) that it v r as impossible 
to keep the cylinders steady if we approached too near to the limit of stability. The 
theoretical limit is X=1’57Y, but in practice X should not be >1 - 25Y. Hence, in 
general, the length of the cylinders has been one-and-a-quarter diameters. 
It is easy to apply the formulae to the case of any two similar surfaces of constant 
curvature. 
The unduloid and nodoid are the roulettes of the foci of an ellipse and hyperbola 
respectively. Lindelof has shown that the sum of the principal curvatures at any 
point in the surfaces of revolution described by these curves about their axes is the 
* ‘ Encyclop. Brit.,’Art. “Capillary Action.” 
