1064 
the moment of momentum of the body with respect to these coor- 
dinates. In the point P this system is geodetical at any moment, 
but the quantities g,, will, in contrast to what was the case before, 
depend on the time 2, and will be periodical with respect to the 
F eee, 
time in P with a period 7’=-—, where RF represents the absolute 
R: 
value of the rotation-vector. In the neighbourhood of P we have 
now, instead of (20), the following formulae 
Gus = Ep» +4 (Gusmn)P Em Ems - .. se « (22) 
where the quantities (4, m.)p are periodical functions of the time. 
In order now to determine the variation with the time of the moment 
of momentum round the «,-axis we make again use of the impulse 
equations (19), and get from these: 
oT,’ or 4 
a. ee ae Ee + $ (4%, Juv3 — Cy Joa) Te. 
Integrating again over a closed surface in the «,—a,—.«#,-space, 
which encloses the body, we find with neglect of terms of the order 
av*,xv?,ev*, and higher orders: 
Aas: —(fe. is = Bee) is) = —4 fie, PRB ee 
— #, (Joork)p)ar TO dS. . . ee ree | (3) 
The left side represents the variation with the time woth moment 
of momentum of the body round the w‚-axis; the right side may 
directly be interpreted as the 2,-component of the couple, which a 
field of acceleration with potential + tg, exerts on the body, and 
is obviously closely connected with the integrals fazer TdS, which 
determine the ellipsoid of inertia of the body. In case of a homo- 
geneous spherical body they are as is well known equal to zero. By 
means of (23) and of the two analogous equations, which refer to 
the moment round the «#,-axis and the z,-axis, the motion of the 
body round its centre of gravity in the stationary field of gravita- 
tion may thus be determined completely. It may be described as a 
Poinsot-motion, which is more or less disturbed by the influence of 
a field of acceleration with potential + 4 g,, (right side of (23)), 
and on which is superposed a uniform rotation, the components of the 
angular velocity of which are given by AZ, and R,. The latter rotation 
is quite independent of the properties of the body, in contrast to the 
influence of the field of acceleration, which is intimately connected 
with these properties, and which e.g. disappears, if we have to 
do with a homogeneous spherical body. 
