112 
Proceedings of Royal Society of Edinburgh. [sess. 
From this, — and which will he required below, are 
dx 2 dx^ 
d*r, 
found by using the factors - r 2 and r 4 . 
The stress on the free surface (where y = i 7 , a quantity of the 
order A) is, by (1), 
(#)<>= -W-KSiv.^ + vsy^)o . . (io) 
where, in p 0 , we must include the effects of the tension T, and of the 
flexural-rigidity E, of the surface-film. 
But, by (3) and (5), we have 
d 
\/(p + e P)= -(e— + pV 2 )Vw.k; 
so that as 
we have 
P=gy 
dp + egdy = enfridy - rdx)A & ry+(rx+nt)l . 
From this, by integrating, and introducing the surface conditions, 
H - p 0 - egr, - Tg + E g + enAi *, 
If we now substitute this in (10) and, for the boundary con- 
dition,* make 
d 
o - o > 
dt 
(omitting terms of the second degree in A and B), we have by 
means of (9) the two equations 
R(A + B) - en 2 A + 2prm(rA + sB) = 0 , 
r 2 ( A + B) + r 2 A + s 2 B — 0 , 
where, for shortness, 
R = egr + Tr 3 + Er 5 (11). 
Thus, finally, 
a 2 
^ ~\ A . (12). 
R - en 2 + i[xnr 2 L + 4^-r 3 (r - s) = 0 
This must be treated differently according as p is small or great. 
I. Let p be small ; and let n be given, and real. This is the case 
of the sustained waves in Prof. Smith’s experiments. 
The equation obtained by neglecting p, viz. 
R - en 2 = 0 
* W. Thomson, Camb. and Dublin Math. Journal , iii. 89 (1848). 
