292 
Proceedings of the Royal Society 
3. Note on an Inequality. By Prof. Tait. 
The tendency of the air contained in a soap-buhble is to ex- 
pand in bulk, while that of the film is to contract in surface. 
Hence it becomes a curious question to decide whether both of 
these tendencies are not satisfied when two or more bubbles unite 
into one. 
The excess of pressure inside a bubble, of radius r, over the 
pressure of the external air, is 
, 4 T 
P ~ V = — ’ 
where T is the contractile force of the film. 
Hence 
£ - 1 + 
p pr 
Now if r' be the radius of the sphere which the contained air would 
occupy at the pressure p, we have 
'3 
P 
V 
so that 
r 3 + 
4T 
p 
r 2 . 
4 T . 
The quantity — is exceedingly small, so that if we call it 3 e, we 
have very nearly 
r' = r 4 - e. 
That is, the quantity by which the radius of a bubble must be 
increased, so that the pressure of the contained air may fall to that 
of the atmosphere, is independent of the radius of the bubble 
(unless it be very small). 
Suppose two bubbles of radii r x and r 2 to coalesce, and let r 3 be the 
radius of the single bubble formed, we have (by expressing that 
the whole content is unchanged). 
r 3 3 + 3 er.f = r x 3 + r 2 3 + 3 er x 2 + 3 erf 
Hence, if SV express the increase of volume, 8S that of surface 
we have 
SV + eSS - 0. 
