496 KINEMATICS OF MACHINERY.
forms here a higher pair with the thread, and is itself the tool. Kelatively to each other, however, the fibres of the thread act as tools. If we imagine for the sake of simplicity simply a pair of such fibres stretched between the spindle S and the draw-frame D,
S. D.
and the spindle to make half a revolution, there is, in the first place, a mere crossing of the two fibres,
but as the turning continues the fibres twist round one another, each fibre acts as a tool in working the other, the screw form of each being simply its envelope relatively to the other. Thus we see that it is not even absolutely necessary that the tool should be harder than the work-piece, and also that occasionally it is not possible to distinguish one from the other. This, however, does not affect our proposition that the work-piece forms a part, or the whole, of one of the links of the kinematic chain forming the machine.
"We notice further that in the work-piece we have a member common to both place- and form-changing machines. We have already noticed that when the so-called tool of the former dis- appeared, the body on which work was done, the work-piece (where it existed) became a part of the machine. In place-changing machines, therefore, as well as in those which we have been con- sidering in this section, the work-piece is a part or the whole of a kinematic link. In this point the two classes of machines are completely alike.
There follows, lastly, from what we have now found as to the nature of the tool, a proposition which is very important, and which has most numerous applications in mechanical technology. It is the following : in order that a given form may be given to a body by a machine, we must give to the tool of the latter the envelope of that form. In order to determine this envelope the intended motion of the tool relatively to the work- piece must first be fixed, and as this relative motion may be of many different kinds, not only may the problem admit of several solutions, but as a rule it itself includes numerous other problems. In every case, however, it is a matter of very great importance to
