16 
DR. R. S. BALL’S RESEARCHES ON THE DYNAMICS OP A RIGID 
As a vector expresses the entire conception of the movement of a particle from one 
position to another, so a twist expresses all that is involved in the movement of a rigid 
body from one position to another (Chasles). 
As a force expresses the resultant of a number of forces applied to a particle, so a 
wrench expresses the resultant of a number of forces applied to a rigid body (Poinsot). 
If a body receive a twist about the screw A, and then a twist about the screw B, the 
resulting position could have been produced by a twist about a third screw, C. 
The three screws A, B, C lie upon the “ cylindroid,” a conoidal cubic surface of which 
the equation is 
z[x 2 -\-y 2 )— 2mxy = 0 . 
The pitch of the generator which is inclined to the axis of x at the angle 6 is 
^>+mcos 23, 
© 
where p is an arbitrary constant. 
I. ON THE YIRTUAL COEPPICIENT OP A PAIR OP SCREWS. 
1. Definition of the virtual coefficient. — If a body receive a twist about a screw A, 
of pitch a , through a small angle a, while acted upon by a wrench P about the screw B, 
of pitch b , the quantity of energy expended is [art. 18] 
a . P . [(«+£») cos 3 — d sin 3], 
where d is the length of the common perpendicular to A and B, and 3 is the angle 
between A and B. 
Perhaps the simplest rule to distinguish between 3 and its supplement is the following. 
Suppose the common perpendicular to be a screw, in the ordinary sense of the word, and 
that there is a nut on this screw to which A is attached. If, then, the nut be turned so 
as to make A approach B (that is, to make the length of the common perpendicular 
diminish), the angle through which A has turned when it has become parallel to B is 
the angle 3. 
It is a remarkable consequence of the symmetry of this expression, that precisely the 
same quantity of energy is required to twist a body about B, through an angle a, against 
a wrench P about the screw A. 
The quantity within the brackets may be called the virtual coefficient of the pair ot 
screws. In the former paper considerable application was made of the case where the 
virtual coefficient vanished, and the screws were then said to be reciprocal [art. 18]. 
We now proceed to show some results which can be derived from the reciprocal character 
of the expression in cases where the virtual coefficient does not vanish. 
2. Analogy of the composition of rotations to the composition of forces. — It is a matter 
of great interest that angular velocities are compounded like forces, and translations like 
couples. The source of the analogy in the general principles of virtual velocities has 
been traced by Rodeigues (Liouville’s Journal, t. 5, 1840, p. 436). In the former paper 
