a pair of these curves, (one belonging to the one and the other to the other element), be isolated, they can therefore be caused to slide upon one another without destroying their coincidence. Thus if in each of two points A and A' of both curves (Fig. 37), an
Fig. 37.
osculating plane be laid, and if further by its means the homo- logous systems of coordinates xyz, x y z' be drawn, then if A B and A' I? be portions of equal length, and A be brought to A and x to x, B must come to B', and the whole line A B must coincide with A' B'. Among the lines which fulfil this condition we have
first, where extension alone is con- cerned, the straight line; among plane curves, that is curves of two dimensions, we have the circle only, and among general curves of three dimensions, the cylindric helix only. The two first may, however, be considered as special cases of the last, so that we may say that the only curve fulfilling the required condition is the cylindric helix. The common screw and nut therefore form
closed pair (Fig. 38). ^ The form of the screw is not however in this case completely indifferent, on account of the third condition that only a single motion shall be possible, which here must be along the helix itself. Screw and nut must therefore be so shaped that each and every
Fig. 38.
