646 MR E. M. WEDDERBURN 
at the boundary. When there is such a rise, ¢, at the boundary, the change of volume of 
any lamina, A’,(x), above the ihe is — b’,(x)C, ( 1 —), and of any lamina A,(z) 
of the lamin A’,(x) and A,(2). 
The equations of continuity for the various laminz are then of the form 
A’,(@)dar = {A'n(0 - &,) - Vela) n}de( 1 i =) 
and 
A,(z)de= (Ag(x + &) ~ bi @)bu}dx(1 +2) 
That is | 
£'n0'n(a) = - A’n(2t) / Ue — =)+ NCE) (2), 
tabula) = Au(a)|(1+ 3%) + Aa). 
or neglecting quantities of the second order 
£,= - re) ee i , 
be 5 alee 
2 
Neglecting vertical accelerations (7.e. assuming (4) small, and the amplitude of 
the oscillation also small) and considering only horizontal acceleration, the differences 
between the pressures on the two sides of the various lamine in their disturbed 
positions will simply be due to the difference in density between successive lamin and 
the change in their relative levels, eg. for the lamina A’,(«), p’,C, per unit area of 
the whole lamina, where p’, ‘is the difference between the density of that lamina 
and the adjoining lamina, the resultant pressure for 8’ being g=p’,¢’,, per unit area, and 
for 8, gZpnG, per uvit area; g being the acceleration due to gravity. 
The equation of motion for all the lamine, as a whole, then becomes 
i ; ag", : 1 0€, 
roe A (a)da( 1 - £2) (Aa@Ea} + xen Aste) 1+ 2 als (x)é,} 
= - glx(Zp'nb'n + Spaln) ; : (4) 
where R’,, R’,, RB’, . . . R,, Ry, Rg . . . represent the density of successive lamine. 
But if the various lamine are to remain in contact we must have 
H (aedea}= —lAed— oe 
