A property independent of the particular axes of co-ordinates used in the representation of the curve by its equation; for instance, the curve may have a node, and in order to this, a relation, say A = o, must exist between the coefficients of the equation; supposing the axes of co-ordinates altered, so that the equation becomes u' = o, and writing A' = o for the relation between the new coefficients, then the relations A = o, A' = o, as two different expressions of the same geometrical property, must each of them imply the other; this can only be the case when A, A' are functions differing only by a constant factor, or say, when A is an invariant of u.
Siace E2 + if + ~1, or ef, is necessarily an absolute invariant for all transformations of the (rectangular) co-ordinate axes, we infer that XE + un + v~ is also an absolute invariant.
When the latter invariant, but not the former, vanishes, the displacement is equivalent to a pure rotation.
When the latter invariant, but not the former, vanishes, the system reduces to a single force.
This expression gives the work done by a given wrench when the body receives a given infinitely small twist; it must of course be an absolute invariant for all transformations of rectangular axe~.