546 VISCOSITY. [CHAP, xi 



provided V^ = 0, 



(5), 



where V, 2 = d^daf + d"/df . 



To determine the ' normal modes ' which are periodic in 

 respect of a, with a prescribed wave-length 2-7T/&, we assume a 

 time-factor e?* and a space-factor e ikx . The solutions of (5) are 

 then 



with ra 2 = & 2 + a/z/ ........................... (7). 



The boundary-conditions will supply equations which are sufficient 

 to determine the nature of the various modes, and the corre- 

 sponding values of a. 



In the case of infinite depth one of these conditions takes the 

 form that the motion must be finite for y = oo . Excluding for 

 the present the cases where m is pure-imaginary, this requires 

 that B = Q, D = 0, provided m denote that root of (7) which has 

 its real part positive. Hence 



If ?; denote the elevation at the free surface, we must have 

 drj/dt = v. If the origin of y be taken in the undisturbed level, 

 this gives 



< n = --(A-iC)e ikx+ t ............... (9). 



a 



If T l denote the surface-tension, the stress-conditions at the 

 surface are evidently 



Pyy = T ' **-<> .................. (10). 



to the first order, since the inclination of the surface to the 

 horizontal is assumed to be infinitely small. Now 



dv du 



