34 
DE. E. S. BALL’S EESEAECHES IN THE DYNAMICS ON A EIGID 
If, however, the two screws of zero pitch become imaginary on the cylindroid, as will 
he the case with certain dispositions of the constraints, it will not be possible for the 
body to remain in equilibrium, no matter how it may be placed. 
If the body had five degrees of freedom, it can only remain in equilibrium when acted 
upon by a wrench about the single screw reciprocal to the freedom. If the restraints 
were such that the pitch of this screw were zero (which of course will not generally be 
the case), then when the vertical through the centre of inertia coincided with this line 
equilibrium would subsist. In general, however, it is impossible for a body with five 
degrees of freedom to be in equilibrium under the action of gravity. 
On the other hand, if the body had only one degree of freedom, through every point 
in space a plane can be drawn such that every line in the plane passing through the 
point is a direction along which, if the vertical through the centre of inertia acted, 
equilibrium would subsist. 
39. Equilibrium of four forces applied to a rigid body . — If the body be free, the four 
forces must be four wrenches about screws of zero pitch which are members of a 
coordinate system with three degrees of freedom. The forces must therefore be 
generators of an hyperboloid, and all belonging to the same system [art. 81]. The 
relative magnitudes of the four forces P, Q, It, S are easily determined when the posi- 
tions are known. Draw the cylindroids (P, Q) and (E, S), then T, the common screw 
of these cylindroids, makes angles with P and Q, the sines of which angles are in the 
proportion of Q to P. 
Three of the forces, P, Q, E, being given in position, S must then be a generator of the 
hyperboloid determined by P, Q, E. This proof of a well-known theorem is given to 
show the facility with which such results flow from the Theory of Screws. 
Suppose, however, that the body have only five degrees of freedom, we shall find that 
somewhat more latitude exists with reference to the choice of S. Let X be the screw 
reciprocal to the freedom of the body. Then for equilibrium it will only be necessary 
that S be coordinate with the four screws 
P, Q, E, X. 
Now a cone of screws can be drawn through every point in space coordinate with the 
four screws just written, and on that cone one screw of zero pitch can always be found 
[art. 89]. Hence one line can be drawn through every point in space along which S 
might act. 
If the body had only four degrees of freedom, the latitude in the choice of S is still 
greater. Let X„ X 2 be two screws reciprocal to the freedom, then S is only restrained 
by the condition that it be coordinate with the five screws 
P, Q, E, X„ X 2 . 
Any line in space when it receives the proper pitch is a screw coordinate with the five 
screws just written. Through any point in space a plane can be drawn such that every 
