28 
DR. R. S. BALL’S RESEARCHES IN THE DYNAMICS OF A RIGID 
This is thus proved. An impulsive wrench about X can be decomposed into impul- 
sive wrenches about the screws P, B„ &c., B 6 _j. All of these will be destroyed by the 
reactions of the constraints except the first ; but by hypothesis an impulsive wrench about 
P will make the body twist about S. 
For example, if the body had five degrees of freedom, then any impulsive screw on a 
certain cylindroid possesses the property required. 
If the body had three degrees of freedom, then any screw reciprocal to a certain cylin- 
droid would possess the required property. 
26. Definition of principal screws of inertia. — A rigid body has Jc degrees of freedom ; 
it will be shown that from the coordinate system expressing that freedom, k screws may 
be selected such that an impulsive wrench about any one of these screws will make the 
body commence to twist about that screw. 
The k screws possessing this property may be called the principal screws of inertia. 
We shall first demonstrate the existence of six principal screws of inertia in a per- 
fectly free rigid body. Out of the thirty elements which in general determine a set of 
six screws, fifteen are still disposable in a set of six conjugate screws of kinetic energy. 
We are thus enabled to choose one set of conjugate screws of kinetic energy [art. 55] 
which are also coreciprocal (art. 7). The set thus chosen manifestly possesses the pro- 
perty that an impulsive wrench about any one of them makes the body commence to 
twist about that screw. 
It can be shown that this set is in general unique. For let the group be represented 
by Aj &c., A 6 , and let S be another screw assumed to possess the same property. An 
impulsive wrench about S can be decomposed into impulsive wrenches about A! &c., A 6 . 
These wrenches will produce twist velocities about Aj &c., A 6 . If these twist velocities 
could compound into a twist velocity about S, it would follow that equal twist velocities 
should be produced about each of the screws A t &c., A 6 by equal impulsive wrenches. 
If this were the case, then every impulsive screw would be its own instantaneous screw, 
which manifestly is not true. Hence there can be only one set of six principal screws 
of inertia. 
2 7. Construction of the six principal screws of inertia for a free rigid body. — Let O A, 
OB, OC be the three principal axes through the centre of inertia of the rigid body, 
and let a , b, c be the corresponding radii of gyration. Let two screws, A', A", be 
formed along OA with pitches -\-a, —a, and let two other pairs of screws, B', B" and 
C', C", be similarly' formed along the other principal axes, OB and OC. The six screws, 
A', A", B', B", C', C", are the set required. 
28. Principal screws of inertia of a constrained rigid body. — Of the k {k— 1) arbitrary 
elements in the selection of any k screws from a coordinate system of freedom k, half 
the entire number still remain disposable in k conjugate screws of kinetic energy 
[art. 87]. The remaining disposable elements are just the number of conditions required 
that each of k screws should be reciprocal to all the rest. We have therefore just 
sufficient latitude for the determination of k conjugate screws of kinetic energy, which 
are also coreciprocal. We shall now show that the k screws thus defined possess the 
