general case of the twisting of axoids, reduced at the same time to the most simple imaginable form, where both axoids are concentrated in the twisting axes themselves.
With the turning-pair we can observe something very similar to this. Here all points in the moving eye, the open ele- ment, describe circles about points in the geometrical axis of the stationary cylinder, these circles being equal for points at equal distances from the axis. The axoid of the fixed body is thus again a straight line coinciding with its geometrical axis, and we find the axoid of the eye to be the same, if we fix it and cause the cylinder to move. Thus for the axoids of this pair of elements, the full and open revolute, we have again two coincident axes turning about each other, forming the simplest case of cylindric rolling which we can conceive, one in which both the cylinders of instantaneous axes have become merely straight lines.
With the prism-pair all rotation ceases; the twisting of the instantaneous axes becomes a simple sliding of them one along the other. The geometrical axes of both prisms may be considered to be their axoids, but the notion of the geo- metrical axis is not so determinate in the prism as in the cylinder or screw; and we can consider any given pair of coincident edges or parallels to edges to be axoids.
Here therefore the other extreme of the most general case of twisting is realized that in which the sliding alone remains.
We may now advance a small but important step. We have i above laid down as the first condition for the attainment of a given motion by one pair of bodies that one of the elements must be rigidly connected with the portion of space which we have considered as stationary. We may now release ourselves from this condition. For if two elements which have been rightly paired ' be both set in motion, there still continues between the one element and its partner the former absolute motion, (or motion which we have agreed to consider absolute,) but it has now become the relative motion of the element to its partner. We may there- fore arrange the pairs of elements which we have found in kine- matic chains, where then the relative motion of the paired elements becomes that also of the links which they connect.
