1063 
that the influence of the centrifugal force may be disregarded. This 
simplification can also be expressed by saying that in the integration 
of the general differential equations all terms are left out which are 
of a higher order of magnitude than the first with respect to the 
velocity which is treated as infinitely small). From a theoretical 
point of view there is no objection to this supposition, but where 
the object is to use the theoretical results in the experimental deter- 
mination of viscosities of liquids or gases, it is necessary to know 
for a given liquid the limit of velocity (or rather of amplitude of 
oscillation, which limit may also change with the time of swing) 
beyond which the theory is no longer applicable in practice: in 
other words what the error is with a given amplitude caused by 
the simplifying supposition. 
The results of the approximate theory were used in the deter- 
mination of the viscosities of mixtures of oxygen and nitrogen’). 
The velocities occurring in these experiments (not higher than .04 
cm. p. sec.) would seem to be sufficiently small and a definite 
indication, that the deviation from the simplified theory could not 
be considerable in these experiments, was given by the fact, that 
over a fairly large range of angles of deviation (between 4° and 1.5°) 
the logarithmic decrement d of the amplitudes appeared to be very 
nearly independent of the amplitude itself, whereas the opposite 
might be expected, if a deviation from the theory existed. Moreover 
the method, when applied to water, appeared to yield satisfactory 
results ®). 
A perfectly trustworthy proof, that the velocities could actually 
be looked upon as small, was, however, not available. From a 
consideration of the order of magnitude of the terms neglected in 
the differential equation one might even be inclined to conclude 
that such was not the case. Indeed a development of the equation 
shows that neglecting the terms of the second order (us ete. 
U 
, Ou 
as compared to the terms of the first order such as 5,” comes to the 
Ow 
same as neglecting w’* with respect to aa (w being the angular velo- 
city of the oscillating body) and this leads to the simple conclusion, 
that the approximate theory is only applicable, if the angle of 
1) Comp. e.g. G. Kircunorr, Vorlesungen über mathematische Physik, n°. 26. 
2) Comm. N°. 1495. Proc. XVIII, 2, p. 1659. 
3) Comp. for instance Comm. N°. 1485, § 16. 
