If to increase the area of the surface requires the expenditure of work, 

 the Mir&ee roust reewt extension, and if the bubble in contracting can do work, 

 the Kirface roost tend to contract The surface must therefore act like a sheet 

 of india-rubber when extended both in length and breadth, that is, it must 

 exert uriace-tension. The tension of the sheet of india-rubber, however, de- 

 pend* on the extent to which it is stretched, and may be different in different 

 directions, whereas the tension of the surface of a liquid remains the same 

 however much the film is extended, and the tension at any point is the same 

 in all directions. 



The intensity of this surface-tension is measured by the stress which it 

 exerts across a line of unit length. Let us measure it in the case of the 

 spherical soap-bubble by considering the stress exerted by one hemisphere of 

 the bubble on the other, across the circumference of a great circle. This stress 

 is balanced by the pressure p acting over the area of the same great circle : 

 it is therefore equal to irr*p. To determine the intensity of the surface-tension 

 we have to divide this quantity by the length of the line across which it acts, 

 which is in this case the circumference of a great circle Zirr. Dividing irfp 

 by this length we obtain %pr as the value of the intensity of the surface- 

 tension, and it is plain from equation 8 that this is equal to T. Hence the 

 numerical value of the intensity of the surface-tension is equal to the numerical 

 value of the surface-energy per unit of surface. We must remember that since 

 the film has two surfaces the surface-tension of the film is double the tension 

 of the surface of the liquid of which it is formed. 



To determine the relation between the surface-tension and 

 the pressure which balances it when the form of the surface 

 is not spherical, let us consider the following case : 



Let fig. 8 represent a section through the axis Cc of a 

 soap-bubble in the form of a figure of revolution bounded by 

 two circular disks AB and ab, and having the meridian section 

 APa. Let PQ be an imaginary section normal to the axis. 

 Let the radius of this section PR be y, and let PT, the 

 tangent at P, make an angle a with the axis. 



Let us consider the stresses which are exerted across this 

 imaginary section by the lower part on the upper part. If 

 the internal pressure exceeds the external pressure by p, 

 there is in the first place a force infp acting upwards arising 



