234 Proceedings of the Royal Irish Academy. 
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 conoidal cubic surface, 
whose equation is 
z {x^ + - 2 mxy = 0 
this surface has been named the cylindroid, at the suggestion of Pro- 
fessor Cayley, 
All the generators of the cylindroid are screws. The pitch of the 
generator which is inclined to the axis of x at the angle 0 is 
p + m cos 2 0 
where ^ is an arbitrary constant. 
The cylindroid is completely determined when two of its screws 
are given. 
All cylindroids are similar surfaces, depending upon a single para- 
meter m. Two parallel planes at a distance 2 m include the entire 
surface between them. 
A model of the cylindroid has been constructed. The generating 
lines are formed by steel wires forced into holes properly placed upon 
a boxwood cylinder the axis of which is the nodal line of the surface. 
The pitch is shown by colouring a length m cos 20 {p = 0) upon 
each generator. Different colours indicate the sign of the pitch whether 
+ or -. 
The fundamental property of the cylindroid is thus stated : If 
three screws upon the surface be taken, and if the body be twisted 
about each screw through a small angle proportional to the sine of the 
angle between the other two, the body after the last twist will occupy 
the same position as it did before the first. 
A wrench about a screw A^ and a wrench about a screw B, act 
upon a body. Determine the single screw C, a wrench about which 
shall be equivalent to the wrenches about A and B, acting together. 
The screw C lies upon the cylindroid, which is determined by 
A and B. Three wrenches about three screws upon a cylindroid will 
make equilibrium (when applied to a rigid body), if each of the 
wrenching forces be proportional to the sine of the angle between the 
two remaining screws. This theorem includes (jt? = 0, w = 0), the 
ordinary law for the composition of forces, and = oo , m = 0), the 
ordinary law for the composition of couples. 
Poinsot has shown that the composition of rotations is analogous to 
the composition of forces, and that the composition of translations is 
analogous to the composition of couples. These analogies are now 
shown to arise from the identity of the rules for compounding twists 
and wrenches by the cylindroid. 
It is also to be remarked that, so far as the composition of forces, 
couples, rotations, or translations, are concerned, the plane can only 
