1076 
Kee (ow)? 
A Be 
These equations are satisfied by putting: 
== Ik by, EN SP eee ae EN 
(9) 
where w,, Y,, 7, are new functions of g,z, and 6. A motion of cir- 
culation is thus obtained in the meridian planes; this motion is a 
damped pulsating one, with twice as high a frequency and degree 
of damping, as the oscillation in first approximation. 
5. In third approvimation we have again Ap, =0, or p, =0, and 
\ 
Es rete: 
n À u, Fuut 
where this time 
KD ed er Den (11) 
with 
0g, Op, 
Db, 2h a EEN? EN ak tn). 
In a third approximation we thus have a motion caused by a 
periodic damped field of force at right angles to the meridian planes 
and containing the time in the factor e®*‘. It follows, that this motion 
like the one in first approximation consists of an oscillatory rotation 
of the liquid in shells, each with its own amplitude and phase, but 
with the same period and degree of damping; i.e. the equations 
can be satisfied by putting: 
DS 
p‚ being a new function of 0, z, and 5, determined by the differen- 
tial equation : 
re Le Ee ing, = 20+ (2.0, + on tg) (0 
dv? Hig do 7 z 
and by the nn that p, =O at the boundaries of the liquid. 
6. As one would be inclined to expect and as, moreover, can be 
easily proved, further approximations yield alternately circulation 
in meridian planes and oscillations, about the axis, with frequencies 
and degrees of damping which increase in an arithmetical series. 
By putting 
— X, = Ge Dy ea a = Se Dr 
one finds, using a well known method *) 
1) If the relations hold for n= 1 to m, it may be proved, that they also hold 
for n=m-+1. 
