1038 
Physics. — “The viscosity of liquefied gases. UI. The aperiodic 
rotational motion of a sphere in a viscous liquid.” By Prof. J. E. 
VERSCHAFFELT. (Comm. N°. 148d from the Physical Laboratory 
at Leiden). (Communicated by Prof. H. Kamenincu ONNms), 
(Communicated in the meeting of November 27, 1915). 
. 
1. In my previous communication (Comm. 1485) on the rotational 
motion of a sphere in a viscous liquid the motion was supposed to 
be periodic’). The question as to whether the motion can be aperiodic 
and, if so, what conditions the periodicity of the motion is subject 
to, did not need to be raised for the experimental object of the 
theoretical investigation. For the sake of completeness and with a 
view to the possibility of determining the viscosity of a liquid by 
the observation of an aperiodic motion, we here give the solution 
of the case in question. 
It may be seen at once, that aperiodicity is out of the question 
in the case of a sphere oscillating in an infinitely extended liquid. In 
the discussion of the similarity of the motions (Comm. 148c) it appeared, 
that in every liquid to each set of values of A (moment of inertia 
of the oscillating system) and J/ (moment of the rotational couple) 
a definite set of values of 7’ (time of swing) and d (logarithmic 
decrement of the amplitude per time of swing) corresponds; 7’ never 
becomes infinite, the necessary condition for the transition from a 
periodic to an aperiodic motion, unless in the limit, when J/ becomes 
zero. But in that case 6" as well as 6’ become zero, therefore / is 
1) It seems advisable to emphasize the fact, that theoretically our equations are 
only valid for infinitely small velocities; this was possibly not stated with sufficient 
distinctness in the previous communication. It is only with small velocities that 
a motion of the liquid in spherical shells can be assumed, because in that case 
the centrifugal action which is proportional to the square of the angular velocity 
may be disregarded. With a constant angular velocity the centrifugal force would 
everywhere be neutralized by pressure-differences, but in our case, where w de- 
pends on 7, this is not the case, and the rotating sphere acts on the liquid as a 
centrifugal pump (vid. Stokes, Math. and Phys. Papers, 1, p. 88), in such a manner, 
that a circulation of the liquid arises from the equator towards the poles. This 
is easily seen by considering, that the centrifugal force points everywhere away 
from the axis and is zero over the external boundary and along the axis so that 
the circulation of that force round the closed path: equatorial radius—quadrant 
of meridional circle R’—axis—quadrant of meridional circle Rf, is positive. How 
great, practically, the influence of the centrifugal action is with small angular 
velocities, would have to be specially investigated (comp. also RAyLEraH, Phil, 
_ Mag. (4), 36, 354, 1893; Sc. Papers, IV, 78; Lamp, Hydrodynamics, p. 547—48 ; 
G. ZeMPLÈN, Ann. d. Phys., 38, 84, 1912.) 
