2i NILS ZREILON, 
density and horizontal velocity. The most general conception is that 
of a liquid in a state of motion with density and velocity arranged 
in horizontal strata aecording to arbitrary laws of vertical variation, 
This general problem, whose outlines will be given presently, is related 
to important stability and other questions, but it must be remarked 
that our treatment is so far not very rigorous, as friction is ne- 
cessarily supposed to account for the original distribution of velocity, 
though it is afterwards neglected when treating of the superposed small 
wave-disturbances. 
2. In order to obtain the laws of wave-propagation for any 
fluid horizontally stratified in the above sense, we regard with Burn- 
SIDE the actual fluid as the limit of a great number of horizontal strata, 
within each of which density and velocity are constant. We take the 
r^ stratum, animated by the velocity U, along the (horizontal) axis of 
x, and if A is the thickness of any stratum, 
yeh wird 
are respectively the upper and lower boundaries of the stratum con- 
sidered. Let  denote the vertical displacement of the boundary rA, 
then for y — rh: 
Mer auod 
E at "gu. v WEE 
0i; zl (UF : 07) Ka n Dr 1 
dt fur Quy 07 
where 4» and 4»,, are velocity potentials within the 7" and (r-4-1)^ 
strata, and are given by expressions of the form: 
2) d, = — U,x + &—” (A, cosh ky + B, sinh ky) 
= — U,x+ 9,; A, , BD, constants, 
" 
9 2 
= 
: : 7t ; AT 
supposing waves of length n and period ~~. Denote by the symbol 
; ; : 
4 the difference between corresponding functions of indices r + 1 and 
r, then we have from 1): 
