Page:The Kinematics of Machinery.djvu/290

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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,