of Viscous Liquid under Capillary Force. 147 



u = d<f>ldrj w = d<j>/dz (2) 



If the resistance is /// times the velocity, the general equation 

 of pressure, viz. 



p/p = n-d(f>/dt-iW, (3) 



becomes for the present purpose, where U 2 may be neglected, 

 p=—/jL / <j)—pd(f)/dt (4) 



The quantities defining the motion are as functions of z pro- 

 portional to e ikz , and as functions of t proportional to e int , 

 where k is real, but n may be complex. The general equation 

 for the velocity-potential of an incompressible fluid, viz. 

 V 2 <£ = 0, thus becomes 



dr 2 r dr r 



of which the solution, subject to the conditions to be imposed 

 when r = 0, is 



^ = AJ (ikr), 

 or rather 



= A «*<»*+**> J (/&•) (5) 



At the same time p is given by 



p=-(ji/ + inp)$ (6) 



We have now to consider the boundary condition, applicable 

 when r — a. The displacement £ at the surface is connected 

 with </> by the equation 



H«*=$*-h& (?) 



The variable part of the pressure is due to the tension T, 

 which is supposed to be constant, as is practically the case in 

 the absence of surface-contamination. The curvature in the 

 plane of the axis is —d 2 £/dz 2 , or Jc 2 %. The curvature in the 

 perpendicular direction is (a + £) _1 , or 1/a — ^/a 2 . Thus 



p= — ^ — > < 8 > 



and the boundary condition is 



T(/j 2 a 2 -l)# , , . XJ 



ina 1 dr vr ^ r/r> 



or by (5), 



J? J« +«(m+ / ./ P ) = > . (9) 



a quadratic equation by which n is determined. 



L 2 



