1063 
d » 
aa ke is) a 1 (osn f Uh [oo dS ao (oep fret dS +e 
Lo ' 
= (oare f a TAS +... OEY sp. Depa) 
Here we have omitted terms, which would be of the order v? 
(terms with 7%) and of the order x’. The left side of (21) represents, 
apart from the sign, the variation with the time of the total 
momentum of the body in the direction of the wi-axis. The integral 
in the first term on the right side (order of magnitude x) represents 
the total moment of the body with respect to a plane through P 
perpendicular to the «k-axis and is eqnal to zero, because we have 
assumed that P coincides with the centre of gravity. The integral 
in the second term on the right side (order of magnitude v) is equal 
to the total momentum of the body in the direction of the a,-axis 
and is also equal to zero, because we have assumed that the centre 
of gravity was at rest at the moment under consideration; finally 
the third term is a small term of the order of magnitude zv and 
may be neglected, since we already have omitted terms of the order 
of magnitude #? and v°. From this we see that in first approximation 
the momentum of the body remains zero in the course of time, and 
that consequently also the centre of gravity remains at rest. (Here 
it may be of interest to mention, that it is impossible to fix the 
centre of gravity of a body in an invariant way; if we try to keep 
to the classical definition, there always exists a small uncertainty 
in the position of the centre of gravity, the order of magnitude of 
which may be easily indicated). If we assume that the equilibrium 
of the body in P is stable, equation (21) allows us also to calculate 
the small oscillations, which the centre of gravity can perform in the 
neighbourhood of P, but we shall not enter further into this point. 
We will now proceed to consider the possible motion, characterised 
by three degrees of freedom, of the rigid body round its centre of 
gravity. This may be done most easily by calculating the rate of varia- 
tion with the time of the moments of momentum of a body round 
the axes of coordinates. For this purpose it will be of advantage to 
introduce the system of coordinates, which was discussed at the 
end of the first $, and which appears by the “rotation-transfor- 
mation” mentioned there (see p. 1056). This new system of coordi- 
nates rotates uniformly round tbe point ? with respect to a system 
of coordinates, which is at rest in the stationary teld of gravitation, 
with an angular velocity, the components of which coincide with 
the components /?;, of the “rotation-vector’, and we shall investigate 
69* 
