Ball — On Smalt Oscillations of a Rigid Body. 237 
C, whose pitch is -p. Strictly speaking, the case of freedom VI. is 
the reciprocal of the case where the body is immovable. In the 
latter case the body remains in equilibrium wherever be the screw 
about which a wrench acts. In the former the body can be twisted 
about every screw in space. 
Whatever be the nature of the restraints which prescribe the 
liberty of a rigid body, it is possible to choose a system of K screws 
such that when the body is free to twist about each and all of these 
screws, its freedom is precisely the same as with the given restraints. 
[JS". B. — Small movements only.] 
Any K screws of the co-ordinate system maybe selected to express 
this freedom. 
It is manifest that the power of replacing a given system of re- 
straints by another equivalent system may be useful in the solution of 
mechanical problems. 
One screw can be found in space reciprocal to five given screws in 
space. This is a very important theorem. 
Given a cylindroid and a screw X, one screw Y can always ba 
found, upon the cylindroid such that X and Y are reciprocal. 
Given any seven screws in space. It is possible in one way to 
find seven twists (or wrenches) about these screws, such that their 
effect upon the position (or equilibrium) of a rigid body shall com- 
pletely neutralise each other. 
A construction follows by which we are enabled to decompose a 
given twist (or wrench) about a given screw into six twists (or 
wrenches) about six given screws. 
When a body has ^degrees of freedom, the reaction of the re- 
straints produces a wrench about a screw of the reciprocal system. 
A body having K degrees of freedom is in equilibrium when acted 
upon by a wrench about a screw of the reciprocal system. 
A body which is in equilibrium under the action of any forces 
receives a twist about a screw A (the displacement screw). A wrench 
commences to act about the screw X (the restoration screw). 
The forces which hold a body in equilibrium form a conservative 
system. A, B are a pair of displacement screws; X, F the correspond- 
ing restoration screws. If A be reciprocal to Z, then B is reciprocal 
to X. This appears to be an important property of a conservative 
system. 
A free body is in equilibrium under a conservative system. A set 
of six screws, A^, &c., A^, can be chosen in an infinite number of ways, 
such that each restoration screw, (Xj, corresponding to A^, for ex- 
ample), is reciprocal to the five remaining displacement screws, {Ao, 
&c., Aq.) The six screws are called conjugate screws of potential 
energy. 
If a rigid body receive an impulsive wrench about a given screw, 
the body commences to twist about a certain instantaneous screw. 
X and Y are a pair of impulsive screws, A and B the correspond- 
