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II. Researches in the Dynamics of a Rigid Body by the aid of the Theory of Screws. 
By Robert Stawell Ball, LL.D., Professor of Applied Mathematics and Mechanism 
in the Royal College of Science for Ireland. Communicated by Professor Cayley. 
Received May 29, — Read June 19, 1873. 
Contents. 
Introduction 15 
I. On the Virtual Coefficient of a pair of Screws 16 
II. Coreciprocal Screws 18 
III. The Sexiant 21 
IY. On Impulsive Screws and Instantaneous Screws 24 
Y. The principal Screws of Inertia 27 
YI. Miscellaneous Propositions 32 
INTRODUCTION. 
In a paper communicated to the Royal Irish Academy (“ The Theory of Screws — a geo- 
metrical study of the kinematics, equilibrium, and small oscillations of a rigid body,” 
Transactions of the Royal Irish Academy, vol. xxv. p. 157) the chief features of what the 
writer has ventured to call the Theory of Screws were sketched*. It is the object of the 
present paper to give some further extensions and applications of that theory. The 
chief point which it is now proposed to illustrate is the appropriateness of the method 
to many problems in the dynamics of a rigid body. This will, to some extent, appear 
from the analogy subsisting between the conceptions of the theory and the familiar 
notions to which the conceptions degrade when the rigid body degrades to a particle. 
It should also be remarked that the complete generality of the method with reference 
to forces and constraints gives rise to many theorems of great interest, which could 
hardly be enunciated without the ideas which the theory embodies. 
A screw is a straight line in space with which a definite linear magnitude termed the 
pitch is associated. The pitch may have any value from — co to + co . A body is said 
to receive a twist about a screw when it is rotated about the screw, 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. 
A wrench about a screw consists of a force and a couple : the force is along the screw, 
while the axis of the couple is parallel to the screw ; and the moment of the couple is 
the product of the force and the pitch of the screw. 
* References to the former paper are enclosed in square brackets thus — art. [12]. Reference to the articles 
of the present paper are enclosed in semicircular brackets thus — art. (12). 
