458 Lord Kayleigh on the 



It is not difficult to recognize that this integral must be 

 related to T. For if Q be a point upon the normal equidis- 

 tant with P from the surface A B, the potential at Q due to 

 fluid below A B is the same as the potential at P due to imagi- 

 nary fluid above A B. To each of these add the potential of 

 the lower fluid at P. Then the sum of the potentials at P and 

 Q due to the lower fluid is equal to the potential at P due to 

 both fluids, that is to the constant 2K. The deficiency of 

 potential at a point P near the plane surface of a fluid, as 

 compared with the potential in the interior, is thus the same 

 as the potential at an external point Q, equidistant from the 



surface. Now it is evident that \YqcIz integrated upwards 

 along the normal represents the work per unit of area that 

 would be required to separate a continuous fluid of unit den- 

 sity along the plane A B and to remove the parts beyond the 

 sphere of influence, that is, according to the principle of 

 Dupre, 2T. We conclude that the deficiency in ^V-pdz, 

 integrated along the normal inwards, is also 2T ; or that 



r 



V P <fe=2K.*-2T, .... (38) 



z being large enough to include the whole of the superficial 

 stratum. The pressure p at any point P is given by 



p=V P -K, 



so that 



pdz = 

 Jo 



K.z-2T (39) 



We may thus regard 2T as measuring the total deficiency of 

 pressure in the superficial stratum. 



The argument here employed is of course perfectly satis- 

 factory ; but it is also instructive to investigate the question 

 directly, without the aid of the idea of superficial tension, or 

 energy, and this is easily done. 



In polar coordinates the potential at any point P is ex- 

 pressed by 



the integrations extending over the whole space A C B (fig. 3) . 

 If the distance E P, that is z, exceed the range of the forces, 

 every sphere of radius /, under consideration, is complete, 

 and Vp=2K. But in the integration with respect to z 

 incomplete spheres have to be considered, such as that shown 



