| L − yZ + zY | = | M − zX + xZ | = | N − xY + yX |
| X | Y | Z |
(9)
These are the equations of the central axis. Since the moment of the resultant couple is now
| G′ = | X | L′ + | Y | M′ + | Z | N′ = | LX + MY + NZ | , |
| R | R | R | R |
(10)
the pitch of the equivalent wrench is
(LX + MY + NZ) / (X2 + Y2 + Z2).
It appears that X2 + Y2 + Z2 and LX + MY + NZ are absolute invariants (cf. § 7). When the latter invariant, but not the former, vanishes, the system reduces to a single force.
The analogy between the mathematical relations of infinitely small displacements on the one hand and those of force-systems on the other enables us immediately to convert any theorem in the one subject into a theorem in the other. For example, we can assert without further proof that any infinitely small displacement may be resolved into two rotations, and that the axis of one of these can be chosen arbitrarily. Again, that wrenches of arbitrary amounts about two given screws compound into a wrench the locus of whose axis is a cylindroid.