244-245] EFFECT OF SURFACE TENSION. 443 



where p, p' are the pressures close to the surface on the two sides, 

 and R!, R? are the principal radii of curvature of the surface, to be 

 reckoned negative when the corresponding centres of curvature lie 

 on the side to which the accent refers. This formula is readily 

 obtained by resolving along the normal the forces acting on a rect- 

 angular element of a superficial film, bounded by lines of curvature ; 

 but it seems unnecessary to give here the proof, which may be found 

 in most recent treatises on Hydrostatics. 



245. The simplest problem we can take, to begin with, is that 

 of waves on a plane surface forming the common boundary of two 

 fluids at rest. 



If the origin be taken in this plane, and the axis of y normal 

 to it, the velocity-potentials corresponding to a simple-harmonic 

 deformation of the common surface may be assumed to be 



< = Ce^ cos kx . cos (at + e), 



where the former equation relates to the side on which y is 

 negative, and the latter to that on which y is positive. For these 

 values satisfy V 2 < = 0, V 2 $' = 0, and make the velocity zero for 

 y = + GO , respectively. 



The corresponding displacement of the surface in the direction 

 of y will be of the type 



77 = a cos kx . sin (o-t + e) .................. (2) ; 



and the conditions that 



drj/dt = dfyjdy = d^'fdy, 

 for y = 0, give 



<ra = -kC = kC' ........................ (3). 



If, for the moment, we ignore gravity, the variable part of the 

 pressure is given by 



To find the pressure-condition at the common surface, we may 

 calculate the forces which act in the direction of y on a strip of 



