578 
ROTATION FOE STABILITY OF 
If the velocity w only had existed, the kinetic energy of the system 
would be \c^w 2 , and the momentum c^w ; where 
c 3S = m(i + £ 7 ), 
and 6’ 33 is the effective inertia of the body in the liquid in the direction 
OCy y being a constant depending on the shape of the body. In an 
elongated body c 33 is obviously less than c u . 
For instance, if the body be a prolate spheroid of semi-axes OC— c, 
and OA = a, it can be proved that 
where 
a — 
A 
A+ C’ 
7 = 
2 A 5 
0 
A 
d\ _ 1 , _ 
(« 2 + A) 2 (<j9 + A)*' “ (c 2 - a 2 )% g 
d\ c 
[a 2 + A) (c 2 + A)t “ a 3 (c 2 - a 2 ) ~ 
c 
c 
+ \/c 2 — « 2 
— V 7 c 2 — « 2 
a 0 3 - « 3 )* Iog 
C + \/<? 2 — & 2 
c — %/ c 2 —- a 2 
If the angular velocities p and q only had existed, the kinetic energy 
of the system would be 
O 2 + f) i 
where <? 44 is a constant, called the effective moment of inertia of the 
body in the liquid about an equatoreal axis, and is equal to the couple 
which would have to. act round an equatoreal axis of the body for the 
unit of time to generate the unit of angular velocity; also the com¬ 
ponents of angular momentum about the axes OA and OB would be 
c u p and Cucp 
In the prolate spheroid it can be proved that 
= (a 3 + c 2 ) jf 
C — A 
ry /U _ 
0-A+ c a (0+ 2 A)- 
c z + a* 
If the angular velocity r only had existed, then, since this motion of 
the body will not set the surrounding liquid in motion, the kinetic 
energy would be \c && r 2 ; where c G6 is the moment of inertia of the 
body about its axis of figure OC ; <? 66 = M§a 2 . 
Now, if T denote the kinetic energy of the system when all the 
velocities u, v, to, p, q, r co-exist, T will be equal to the sum of all the 
separate kinetic energies due to each velocity separately, and therefore 
T= £ + c n i ? 2 + c 3 S w 2 + c 44 p 2 + c u q 2 + c 66 r 2 ). 
We must now express % v, w, p, q, r, which are estimated with 
respect to axes fixed in the body, in terms of co-ordinates with respect 
to lines fixed in space. 
