440 KINEMATICS OF MACHINERY.
The last-named piece also consists here of two elements, namely, the above-mentioned prism, paired with that upon b b v and this re volute having its parallel to that of the prism.
The result of our examination is, therefore, that this screw fasten- ing is a kinematic chain of three links, formed from the pairs ($), (H) and (P). If we write its formula in full, disregarding (for the sake of simplicity) the incompleteness of (E) and further replacing (72) by (0) as we know to be possible from 57, it will run as follows :
a b . c
c- ... j..r&i s+ ... if... P^ P+ ... if .~c+.
which we may also write, inverting the lower pairs, and noticing that here there is no difference between | and ||; a b c
and in this we recognise a chain, Fig. 291, which we have already examined. We may use (S f P r G*) for its contracted formula. If we consider the link b as fixed, and a as the driving-link, the special formula of the mechanism is (& P' C')^.
In the applications of the screw-pair to cause rectilinear motion, as in the lathe, or to exert pressure, as in the screw-press, these three links are very distinct, arranged in the first case as (S'P'C'fi, in the latter mostly as (ffP ') The form of cnaiu shown in Fig. 291 is also very frequently met with in screwed joints, as, for example j in the "tapped bolt" or " set-screw " of Fig. 292. We also find various methods used in joining b and b v as, e.g., the key shown in Fig. 293. In screwed joints, however, of whatever form, we always find that the pair SiS~ occurs as part of the chain (S' P' C').
The action of this chain in different cases varies very much. In the screw-press or the screw-cutting lathe, with which in certain re- spects the screw-joint might be compared, it is simply used like any other kinematic chain. In the screw-fastening this is also, strictly speaking, the case, but only within such very narrow limits as are allowed by the compressibility of the pieces b and c, beyond these limits it is not used kinematically. When the machine itself is complete, the screw-joint is no longer used as a kinematic chain ; it therefore does not appear in the kinematic formula of the machine. It has been employed as a chain for a temporary purpose
