268 KINEMATICS OF MACHINERY.

The meaning of this formula is exactly the same as that of the last. The part of it within the brackets, however, shows itself at once to be a simple kinematic chain, consisting of the wheels c and d and their connecting link 2, 3. If we use for this the con- tracted symbols of 59, and contract also the symbol for the pair a, b, we have :

1 a,l c,d,3,2 1

C+ ... | ...G't ... | ... C+C... || ... C~

In some cases it will be quite sufficient to write this in a still shorter form extending the use of our former method of contrac- tion, as ((7+ C') ; or generally for an %-fold train of spur-gearing (C+C%+i), where for n pairs of wheels there are in general n -\- \ axes, or rather turning-pairs (C), required.

6l - Formulse for Chains containing Pressure-organs.

In order to find the formula for a chain which contains a pressure- organ it is frequently advisable to imagine the substitution of a rigid element for the latter the pairing being obviously somewhat altered in consequence and then to transform the formula thus obtained by the re-insertion of the pressure-organ.

In order, for example, to express by a formula the water-wheel (Fig. 188) which we have already looked at, we may first replace the water by a rack with a prismatic guide (Fig. 189) so arranged as to drive the spur-wheel a by its own weight, its action being thus similar to that of the fluid for which it is substituted. The formula will run :

C+ ... | ... C+ 9 P f ... II...P+P- ... -f ... C~

If we now change the link P z ... || ... P + into Q^ ...... Q^

replacing the water for its temporary substitute, we must put V~, the symbol for a vessel of any kind, for P~ t and so obtain as a formula for the water-wheel,

If it require further to be indicated that the channel is un- covered, the water being paired with it, that is, by force-closure,