3 1 o KIN EM A TICS OF MA CH INERT.
Turning again to the fourth method of expansion, 2 within 3, a closer examination of it shows us that the link c may be made with a concentric cylindrical projection, which can be fitted into a cor- responding opening or eye in the link I, Fig. 239. Suppose the mechanism placed on d as before. The coupler b has now become a ring of rectangular cross-section which makes oscillatory motions in the annular groove of the block c. We have in no way altered the mechanism by this, for so long as we keep the pair 3 as a closed turning pair we can alter its profile at will. This con- dition allows us to go still further. Let us suppose the crank a to be the driver, the mechanism having for its special formula ((TJl* 1 -)*, we then have simply the rectilinear reciprocation of the block c to consider. The coupler &, as it is moved to the right, drives c both at A and at D, and as it is moved to the left both at B and at C, we may certainly replace this double contact in each direction by a single one ; and this can be done in several ways. We shall attain the object very conveniently if we substitute a sector of the ring b for the whole of it, choosing the sector so as to include the pin 2, as is shown by the dotted lines. This can then drive the block to the rig! it through A, and to the left through B, the motion of the coupler itself being always an oscillation in the annular ring of c. Of the latter we require to use no more than a piece large enough to afford room on each side of the centre line for the swing of the sector I.
Fig. 240 represents the arrangement altered in this way. It must not be forgotten that I is still the coupler as it was before, and that its motion as a link in the chain remains quite unaltered and completely constrained. Kinematically it consists of just the same parts as before, as does also the link c. The form of the link b is still, C ... || ... C, one of the cylinders being the eye enclosing the crank pin 2, while the other works, with sufficient restraint, in the portion of an open cylinder belonging to c, relatively to which it has exactly its former motion. If we wish to write the links b and c in a manner corresponding to their constructive form we must use the symbol for sector, A, instead of that for the com- plete cylinders in the pair 2, and we thus have :
b c
