Page:The Kinematics of Machinery.djvu/102

 being a simple one. Poinsot proposed that the fixed body (or body assumed to be stationary relatively to the observer) should be imagined to be a screw, and the moving body a nut, in which case the sliding would take place along, and the rotation about, the axis of the screw, as above supposed. But as the motion takes place with variable velocity, both as regards the sliding and the rotation, the angle of the thread in the screw and the nut must be imagined to be continually changing. It is difficult to realize this distinctly: a form so irregular is no longer a body, it cannot be realized even with the most determined effort, indeed, things of so varying a nature as are these screws and nuts can scarcely be grasped better than the abstract idea of rotation and translation in space. 11

Belanger makes two proposals. The first is to imagine a pair of bodies having conic rolling (as above, 11), in which both cones have a motion of translation in space. The a rotation then takes place through the conic rolling, and the sliding through the translation of the pair of bodies. By this means the motion may certainly be realized, but only by using three bodies to find the relative motions of two, a thing which is in certain cases advisable, and even necessary ( 9), but which can only be justified when no simpler method is equally satisfactory.

Belanger's second proposition is to consider the consecutive positions of the axis as forming a pair of ruled surfaces, one for each body, so that the motion is reduced to a rolling of the two ruled surfaces upon each other, with a simultaneous endlong sliding upon each other of the generators which are in contact. Other later inves- tigations have attached themselves to this method of representation. Indeed it follows, as a direct conclusion from what we found above, that the consecutive positions of the instantaneous axes of turning and sliding in each of the two bodies enclose such ruled-surface forms as solids of instantaneous axes. 12

The special motion in which translation or sliding along a straight line and turning about it take place simultaneously is called twisting. We may now also, as we have arrived at the most general standpoint, indicate with a common name the bodies we have found, which, by their motions upon each other, determine the relative motions of the bodies with which they are connected. As the surfaces of these solids are always the loci of a -series of axes, they may be called axoids. What we have