If its tension is T the work required to effect this increase of surface will be TdS, and the energy of the film will be increased by this amount.
If the bubble is in the form of a sphere of radius r this material surface will have an area S = 41rr 2 (I) If T be the energy corresponding to unit of area of the film the surface-energy of the whole bubble will be ST = 41rr 2 T (2) The increment of this energy corresponding to an increase of the radius from r to r-+dr is therefore TdS = 81rrTdr (3) Now this increase of energy was obtained by forcing in air at a pressure greater than the atmospheric pressure, and thus increasing the volume of the bubble.
Hence the equation of work and energy is p dV = Tds (6) 41rpr 2 dr = 8zrrdrT (7) p = 2T/r (8) This, therefore, is the excess of the pressure of the air within the bubble over that of the external air, and it is due to the action of the inner and outer surfaces of the bubble.
Hence TdS = dE = Sde d-edS.
Porphyry says of Origen, Kara Tds rrepi lrpay f caTWV Kai Belot) bo s as `EXX vt cav (Euseb.