found may then be collectively expressed in the following law: All relative motions of two bodies may be considered as the twisting or rolling of ruled surfaces or axoids. The laws already found separately must follow from this general proposition by a diminution and simplification of its conditions. There is indeed no difficulty in realizing the transformation of a ruled surface into a cone or a cylinder (of any form) where instead of the twisting motion only rolling occurs. Nevertheless the con- clusion must not be drawn from this that a pure rolling motion
Fig. 32.
Fig. 33.
can be ascribed only to these two cases (as has been generally done so far as I know). The condition determining the absence of the sliding of the edges or generators is neither that all the axes should intersect in one point, as in the cone, nor that they should be parallel, as in the cylinder, but the higher condition that the two ruled surfaces should be so formed that their infinitely near generators in homologous positions include surfaces of the same figure, or, in geometrical language, that they can be developed upon one another.
