Ball — On Small Oscillations of a Rigid Body. 233 
XXY. — The Theory op Screws. — Part I. A Geometrical Study of 
THE Kestematics Equilibrifm and Small Oscillations of a Rigid 
Body. Ey Robert Stawell Eall, LL. D., M. E. I. A. [Ab- 
stract.] 
[Read November 13, 1871.] 
The following are definitions of special senses which are attached to 
certain words used in this paper. A screw is a line in space with 
which a definite linear magnitude termed the is associated. The 
pitch may have any length from + oo to - oo . A body is said to receive 
a twist about a screw when it is rotated about the screio through the 
angle of twisty and is at the same time translated parallel to the screw, 
through a distance equal to the product of the pitch and the angle of 
twist expressed in circular measure. If the pitch be zero the twist is a 
pure rotation ; if the pitch be infinite, the twist is a pure translation. 
The motion of a body implied by the word twist is the motion of a nut 
upon an ordinary screw. A ivrench about a screw denotes a force {the 
wrenching force) and a couple (the wrenching couple). The force is 
directed along the screw. The axis of the wrenching couple is parallel to 
the screw, and the moment of the ivrenching couple is the product of the 
wrenching force, and the pitch of the screw. If the pitch be zero, the 
wrench is a force only. If the pitch be infinite, the wrench a couple 
only. 
A body can be transferred from one position in space to any 
other position in space by a twist about a certain screw. Any system 
of forces acting upon a rigid body can be compounded into a wrench 
about a certain screw. These beautiful theorems are due to Poinsot ; 
they are the elementary axioms from which the present memoir has 
been developed. 
The co-ordinates by which the position of a rigid body in space is 
usually determined consist partly of angular, and partly of linear 
magnitudes. In the present method the co-ordinates are all twists. 
Thus the position of a rigid body in space is defined by six twists 
about six given screws. 
The importance of such a system of co-ordinates is manifest, when 
compared with the method used for resolving forces. Any system of 
forces is resolved into six wrenches about six given screws, instead of 
into three forces and three couples. 
In the present memoir small angles of twist only are considered. 
One consequence of this restriction is, that the order in which twists 
about different screws are communicated has no effect upon the re- 
sulting position of the body. 
Whatever be the motion of a. body in space, it is at any instant 
twisting about a certain screw, which we call the instantaneous screw. 
The angular velocity with which the angle of twist changes we call 
the twist velocity. 
R. I. A. PROC. — VOL. I., SER. II., SCIENCE. 2 H 
