49 i KINEMATICS OF MACHINERY.
which we have found it to exist, and the general kinematic prin- ciples underlying its use.
Let us first look at the action of the tool in some familiar machine, say a common lathe with a slide-rest in which a bar is being turned. The chisel is held fast in the tool-holder of the rest and moves parallel to the axis of the spindle, the bar to be turned is made to revolve along with the mandril in such a way that the portions of its surface to come in contact with the chisel move always towards its edge. The relative motion of the chisel to the bar is the common screw motion, occurring exactly as if the turn- ing tool were a part of a common nut S~ of which the bar to be turned is the screw spindle S + . The chisel and bar have therefore the motion of the pair S=S + . This pair does not exist at the com- mencement of the work, but as the lathe moves, the chisel (being harder than the piece to be worked on) cuts away those portions of the bar which do not belong to its own enveloping form, $+, upon the spindle. That part of the bar, therefore, over which the chisel has passed, has necessarily taken the form of the element S + , the chisel itself carrying a small portion of the partner element S~. Essentially therefore they form a twisting pair, S~S + , as may be recognised more readily, perhaps, if we suppose the lathe worked backwards, and the chisel passing again over the surface it has already formed. The restraint between the two elements of the pair is not complete in itself ( 18, &c.), but the lathe is so arranged as to supply the want by chain-closure ( 43). We may notice that the chisel has carried the profile of the nut S~ from the begin- ning, while the bar only received the form $+ while the turning was progressing. The pairing therefore of the elements into the form SlJS + is made as the motion of the machine goes on, and at the end of the operation the two bodies are really formed into such a pair.
We have said that the bar being turned takes the form S+. This is visible enough in roughing out work, where a sharp-pointed chisel is used. In finishing or smoothing work, where the chisel edge is made straight and parallel to the axis of the lathe, the bar becomes in external form a cylinder (cf. 15) ; but as regards its pairing with the chisel it is still a screw.
We find in the planing and band-sawing machines the same thing as in the lathe that the tool and the body to be worked on
