394 
Proceedings of the Royal Society of Edinburgh. [Sess. 
sides. Take the origin of co-ordinates at point O, at a distance h above the 
undisturbed level, and draw O X parallel to the canal and 0 Z vertically 
downwards. Let the motion be infinitesimal and let f, be the displace- 
ment components at any time t of any particle of water whose undisturbed 
position is x, z. If the motion of the water be started primarily from rest 
by pressure applied to the free surface, the hydrodynamical equations of 
motion take the well-known form : 
0 
0 
i=2 Z <i>(x,z, t ) ■ 
• • (4); 
from which we obtain by integration 
0 
(= si^ x ’ z ’ t); 
II 
iTl Cl) 
Ok 
• (5)- 
In virtue of the incompressibility of the fluid we have also the equation 
0^ 0JU 
dx 2 dz 2 
( 6 )> 
which shows that if <p be known at all points of the free surface, and if it be 
zero at points infinitely distant, its value can be determined at all points 
throughout the fluid. The equation to be fulfilled at the free surface, 
according to a result given by Cauchy and Poisson, is 
( d A\ 
\dz) z =h g\dt 2 ) z =h 
where g denotes gravity. 
§ 3. Remembering now that every function derived from <p by differentia- 
tions or integrations with respect to x, z, t, satisfies all the equations satisfied 
by the function we see that we have 
(8) 
0 2 </>_ 1 0 4 0 
'dz 2 g 2 ~dt^ 
at a free, or level, liquid surface. Equations (6) and (8) combined give 
us the relation 
dx 2 g 2 dt 4 ’ 
1 
and when we put y in place of <•/>, Kb in place of -, and interchange x and t y 
o 
this equation may be written in the form 
279^ __ n 
dP + K 1 3* 4 0 
(3), 
which is the equation already found for flexural vibrations of an elastic 
bar, when terms depending upon the angular motions of the sections of the 
bar may be neglected. Accordingly equation (9) proves that every 
