1059 
zero, as also #, so that properly speaking we are not dealing with 
an aperiodic, but with a stationary motion, albeit one which proceeds 
infinitely slowly. *) 
2. On the other hand the motion may be aperiodic, when the 
liquid has an outside boundary. This was found to be the case in 
a qualitative experiment with glycerine. It is also easily seen theo- 
retically in the extreme case, where a sphere swings inside a second 
sphere, which is only very little larger (section 19 of the previous 
communication); for that case we found that 
R 
L=§ark'*y RER . . . . . . . (41) 
so that L is independent of the period, and by the relation 
Ké+DTk+M=0....... (26) 
k is only imaginary, i.e. the motion is periodic, when 
en R' 
L<4VK or VMKD> in rp > (83) 
If for instance R=10 and &' —R=.1, with water (y = .01) 
as the liquid, the motion will no longer be periodic, when MA is 
smaller than 16.10°. 
3. A general survey of the various possibilities in this extreme 
case is given by the accompanying graphic representation (fig. J), 
which is of the same nature as that which was given in Comm. 
148c for an infinitely extended liquid. 
K M 
The quantities «= 7 and y= power taken as the parameters 
4 
characterising the condition of motion; the periodic time is given 
by svete 
y 
As long as the motion is periodic, we have, owing to L" = 0 
(comp. equation 28) 
(PETE ets eee ie Aen! 
== = =—. hig alot 
dig Az? TS On eae 2: a ey 
By the aid of these equations the curves 7 =a and 7’= 22 
were drawn (shown as continuous curves; they are asymptotic to 
1) The circumstance that in the case of an infinitely extended liquid the motion 
cannot be aperiodic, also follows at once from the fact, that L” (equation (30) 
of the previous communication) cannot be zero (comp. further down section 5), 
when b'= 0, unless b” = 0. 
67 
Proceedings Royal Acad. Amsterdam. Vol. XVIII. 
