122 Preston— On the Continuity of Isothermal Transformation, &c. 
and if = be taken to represent the normal saturated vapour pressure, that is the 
pressure of a saturated vapour in contact with a plane surface of its own liquid, 
then the saturated vapour pressure in contact with a concave spherical surface, 
of radius 7, is easily shown to be 
“t 27 pp 
2B) —) 0 ae 
Fp ps 
where p, is the density of the liquid, and p, the density of the saturated vapour. 
Hence, the relation (1) connecting p and 7 becomes 
itl! Pi 
a = p+ a i ane (2) 
In this equation, all the quantities other than p and 7 may be taken as remaining 
constant during an isothermal transformation, and consequently, within certain 
limits, the volume and external pressure of the mass should increase together. 
This equation, however, cannot be expected to hold in the extreme case, in 
which the bubble is so small that the mass within it ceases to possess distinctly 
the properties of a vapour, or to the other extreme case, in which the bubbles 
become so large and numerous that the remaining liquid, by reason of bemg 
drawn out into thin films, or otherwise, ceases to behave as a liquid in regard to 
the transmission of hydrostatic pressure, etc. Within certain limits, however, 
equation (2) gives the relation between the external pressure and the volume of 
the mass. 
Thus, in the case of a single bubble, if the whole mass be taken as unity, 
and the mass of vapour within the bubble be m, then the mass of the liquid 
portion will be 1—m, and the whole volume will be 
m 1—m 
= —+ 6 3 
UIs (3) 
But, if the radius of the bubble be 7, we have 
m = Spal (4) 
consequently, equation (3) becomes 
4 1 1) 1 
Vv=5 ie — — —— ||a5 5 5 
smer (s+ Os (5) 
or, denoting the specific volumes of the liquid and vapour by 2 and 2, we have, 
from equation (5), 
v—y,= 37 ( - 3) (6) 
7) 
