BODY BY THE AID OE THE THEOEY OF SCEEWS. 
35. Property of the screws of zero pitch on a cylindroid. — The construction just given 
would fail if P lay upon one of the screws of zero pitch. The movements of P must 
then be limited, not to a plane, but to a line. The line is found by drawing a normal 
to the plane passing through P and through the other screw of zero pitch. 
We thus have the following curious property of the lines of zero pitch, viz. that a 
point in the rigid body on the line of zero pitch will commence to move in the same 
direction whatever be the screw on the cylindroid about which the twist is imparted. 
This easily appears otherwise. Appropriate twists about any two screws, A and B, can 
compound into a twist about the screw of zero pitch L, but the twist about L cannot 
disturb a point on L. Therefore a twist about B must be capable of moving a particle 
originally on L back to its position before it was disturbed by A. Therefore the twists 
about B and A must move the particle in the same direction. 
36. Equilibrium of a rigid body having two degrees of freedom and acted upon by 
gravity. — The cylindroid is first to be drawn which expresses the freedom enjoyed by the 
body. It must be remembered that, whatever be the mechanical contrivances by which 
the body is constrained in its movements, so long as the position is completely defined 
by two coordinates all the displacements which it is competent for the body to accept 
could be communicated by twists about the screws of a cylindroid. 
When a body with two degrees of freedom is in equilibrium, the wrench acting upon 
the body must be reciprocal to the cylindroid, but no other condition is required. In 
the case of gravity the wrench reduces to a force, or the wrench may be said to be about 
a screw of zero pitch passing through the centre of inertia of the rigid body. But a 
screw of zero pitch cannot be reciprocal to the cylindroid, unless it intersect both the 
screws of zero pitch on the cylindroid [art. 21]. 
The necessary and sufficient condition of equilibrium is, therefore, that the screws of 
zero pitch on the cylindroid should each intersect the vertical through the centre of inertia. 
37. Equilibrium of a rigid body having three degrees of freedom and acted upon by 
gravity. — The vertical through the centre of inertia must be one of the screws of zero 
pitch belonging to the system reciprocal to the freedom. Draw the pitch hyperboloid 
appropriate to the freedom [art. 88]. Then all one system of generators are the coor- 
dinate screws with zero pitch, while all the other systems of generators are the reciprocal 
screws with zero pitch. 
The necessary and sufficient condition of equilibrium is, therefore, that the vertical 
through the centre of inertia be one of the reciprocal system of generators on the pitch 
hyperboloid. 
38. Equilibrium of a rigid body having four degrees of freedom and acted upon by 
gravity. — The rigid body can then be twisted about every screw reciprocal to a certain 
cylindroid. For the body to be in equilibrium, the wrench which acts upon it must be 
about a screw on this cylindroid. It is therefore necessary that the vertical through the 
centre of inertia of the rigid body coincide with one or other of the two screws of zero 
pitch on the cylindroid reciprocal to tlie freedom of the rigid body. 
MDCCCLXXIV. F 
