546 VISCOSITY. [CHAP, xi 



provided V^ = 0, 



(5), 



where V, 2 = d?lda? + d?jdf . 



To determine the normal modes which are periodic in 

 respect of as, with a prescribed wave-length 2?r/&, we assume a 

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



then 



&amp;lt;f&amp;gt; = (Ad* + Be-^) e ikx+at 



.. 



with 



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 



v mCe m v) e ikx+at ,} 

 ikCtfw) e ikx+at j ...... 



If TJ 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 



r) = -^(A-iC)e ikx+at .................. (9). 



If Tj_ denote the surface-tension, the stress-conditions at the 

 surface are evidently 



(10), 



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

 horizontal is assumed to be infinitely small. Now 



dv du 



