1889-90.] Prof. Tait on Ripples in a Viscous Liquid. 
113 
gives one, and only one, positive value of r, whose value is 
diminished (i.e., the wave-length is increased) alike by surface 
tension and by surface-flexural- rigidity. Call it r 0 , and let 
r = r 0 + pp, 
then by (12), keeping only terms of the first order in /*, 
(ge + 3TVo + 5Er<j)/o 4- inrli = 0 ..... (13). 
Thus p is a pure imaginary, and therefore the viscosity does not 
affect the length of the waves. It makes their amplitude diminish 
as they leave the source. (For the real part of w belongs in this 
case, if we take n as positive, to waves travelling in the negative 
direction along x, and vice versa.) The factor for diminution of 
amplitude per unit distance travelled by the wave is 
This expression gives very curious information as to the relative 
effects of viscosity on the amplitudes of long and short waves, 
when we suppose gravity, surface-tension, or surface-flexural-rigidity, 
alone , to be the cause of the propagation. 
If the waves be started once for all, and allowed to die out, r is 
given and n is to be found. This is the first case treated by Mr 
Basset. If then n = n 0 be found from 
we may put 
eri 2 = E , 
n = n 0 + p.v . 
By (12) we have, keeping only the first power of p ., 
ev = 2r 2 i , 
which coincides with the result given in § 520 of Basset’s Treatise. 
II. Let p, be large. Suppose r to be given, a real positive 
quantity. Then, by (8), we may eliminate n from (12) and obtain 
fl| + ( s 2_ r 2)2 + 4 r 2( s 2_ r 2) + 4 r .3( r _ s ) = 0 . . . (14). 
A 1 
The first term is very small, and the rest has the factor s — r. 
Omit the term which contains this factor twice, and we have 
VOL. XVII. 
9/4/90 
H 
( 15 ). 
