e. 
NILS ZEILON, 
In the present note we confine ourselves to the simplest case when 
U is the same at any depth. Thus in the following treatment we iden- 
tify U with V, where V is the sought for wave-velocity to be deter- 
mined by A) together with the boundary-conditions. As for these 
boundary-conditions we obtain cases of meteorological interest by sup- 
posing the fluid limited by a rigid plane bottom and unlimited upwards 
with its density converging towards zero, whereas in the hydrographic 
case we have an upper free surface of zero pressure. It is known 
however, from Stokes’ laws of boundary wave-motion, that when the 
differences of density involved are small, the vertical motion of this 
upper surface is very inconsiderable. 'lhus no appreciable error is 
committed, if we assume the fluid to be limited by horizontal walls in 
both directions, an arrangement which, by taking the upper boundary 
to be infinitely distant, also covers the meteorological case. 
3. Suppose then a fluid contained in a uniform rectangular 
canal of the total depth A’ + A". Let y = 0 be taken in any convenient 
way within the layer of transition, at a level where there is a con- 
siderable density gradient; we shall have to resolve: 
g d 0 
d da M T 
B) ( dy) a DU EE d e 
dy 
with the boundary conditions 
a(h") = a(— h') = 0. 
Here for shortness A" and h’ are referred to as the depths of the up- 
per and lower fluids respectively. 
The solution arrived at below is confined to the case of a fluid 
in a state of density-distribution slightly differing from that of an ab- 
rupt change of density at y — 0. We are thus led to transform the 
equation B) into such a form as to make Stokes' well-known law of 
propagation immediately result as the first step of an approximate cal- 
culus. By a process applicable also to the more general case A), we 
effect this by replacing B) by a system of integral equations, viz.: 
