Page:The Kinematics of Machinery.djvu/314

 292 KINEMATICS OF MACHINERY.

unless we wish to combine the two expressions in (C"2 C"2)undefined. The converse anti-parallel cranks will be (C"2 C"2)a, if in the same way we omit the exponent c as superfluous, or (C"2 C"2)undefined if we wish to express the fact that the chain placed either on a or on c gives the same mechanism.

The pair-closure has still to be indicated. This will only be necessary if the action of the mechanism extends over the dead points. If the closure exists, and if it be arranged as in Fig. 208, the formula will run (C"2 C"2)d/(p) a.c ; if as in Fig. 209, (C"2 C"2)d/(p) b.d ; where the existence of the pair-closure is denoted by p (see end of 57), while the brackets and the addition of the symbols for the paired links sufficiently indicate the rest. It will frequently, however, be unnecessary specially to indicate the pair-closure, for the maintenance of the anti-parallelism, -the assumption, that is, of the continued validity of the sign Z, -presupposes it. The anti-parallel cranks have here and there been used, but without being recognised; Dübs's locomotive coupling is an instance, and here the ellipses actually serve as profiles for the buffers.*

68 - The Isosceles Crank-train.

We obtain a special case of the chain (C"4) which has very great theoretical interest if we make a=d, b=c, and, as before, a<c. We have already described ( 47) the pair-closure in a mechanism formed from this chain. Figs. 211 and 212 represent the mechanism first without, and then with, the pair-closure. A diagonal joining the points 2 and 4 of the quadrilateral divides it always into two isosceles triangles, for which reason we shall call the train Isosceles. The writing of the chain is easy after the foregoing; the formula must be, -using the symbol for isosceles given in 47 (C"2<=C"2). If the higher pairing of Fig. 212 have to be expressed this becomes (C"2 <= C"2)/(p)a.c . As with the antiparallel crnks, the higher pairings may here be arranged between


 * Dübs and Copestake's patent coupling was illustrated and described in Engineering, vol. xi. p. 318.