1 
114 Proceedings of Royal Society of Edinburgh. [sess. 
This has real positive roots if, and only if, 
/x 2 r 4 > Be , 
and thus, by (8), when this condition is satisfied n is a pure imagin- 
ary, and there can be no oscillation. Of the two roots of (15) we 
must, in consequence of our assumption (that (s - r) 2 is negligible) 
choose that which is nearly equal to r. It might be fancied that, 
as this assumption leads to B = — A very nearly, a new limitation 
would be introduced as regards the magnitude of rj. But we have 
V= - — (A + B)8 |ra+ ”‘>‘ = - r A (\ — 
n v n \ r 2 + s 2 / 
= - A— nearly. 
fir 
The wave-pattern, in this case, does not travel but subsides in 
situ , its amplitude diminishing according to the approximate factor 
g-Ril/2/Ar 2 
Thus, as was to be expected, the subsidence is slower as the friction 
is greater. Also, if gravitation is the sole cause of subsidence, the 
longer waves subside the faster ; while if the main cause be surface- 
tension, or surface-flexural-rigidity, the shorter waves subside the 
faster. 
III. If there be a uniform film of oxide or dust, in separate 
particles which adhere to and move with the surface, we must add 
to the expression for surface-stress in (10) the term 
-Mr) o= 
= - m(jrn(K + B) + im(rA + , 
where m is the surface-density of the film. 
The equations for the elimination of A/B become 
(B + mm 2 - en 2 + 2/xr 2 m) A + (B + mm 2 + 2/xrsm)B = 0 , 
Ur 2 - — )A + (2r* + — -~)b = 0; 
v ix J \ /x fx y 
so that instead of (12) we have 
(e - m(s - r))(B + mm 2 ) = e 2 n 2 - 4 ixr 2 net + i/x 2 r s (s - r) - mn 2 es . 
