CAPILLARY ACTION. 575 



surface, and at the same time we do work in forcing air into it, and thus 

 increase the energy of the bubble. 



That the bubble has energy may be shewn by leaving the end of the tube 

 open. The bubble will contract, forcing the air out, and the current of air 

 blown through the tube may be made to deflect the flame of a candle. If the 

 bubble is in the form of a sphere of radius r this material surface will have 

 an area 



3=4^ ..................................... (1). 



If T be the energy corresponding to unit of area of the film the surface-energy 

 of the whole bubble will be 



ST=irr>T ......................... . .......... (2). 



The increment of this energy corresponding to an increase of the radius from 

 r to r + dr is therefore 



TdS=8irrTdr ................................. (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. 



Let II be the atmospheric pressure and H+p the pressure of the air 

 within the bubble. The volume of the sphere is 



and the increment of volume is 



cZF=4irrtfr .................................... (5). 



Now if we suppose a quantity of air already at the pressure II +p, the work 

 done in forcing it into the bubble is pdV. Hence the equation of work and 

 energy is 



pdV=Tds .................................... (6), 



or 4wpr>dr = SirrdrT ................................. (7), 



or p = 2T ..................................... (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. "We may conceive this pressure to arise from the ten- 

 dency which the bubble has to contract, or in other words from the surface- 

 tension of the bubble. 



