Encyclopaedia Britannica, or, a Dictionary of arts, sciences, and miscellaneous literature : enlarged and improved. Illustrated with nearly six hundred engravings > Volume 20, SUI-ZYM

Water¬
works.
fig. 10.
WAT •[ 672 ]
water in the buckets. Our mill-wrights know well mathematicians
enough, that too great velocity will throw the water
” out of the buckets j but few, if any, know exaftly the
diminution of power produced by this caufe. The fol¬
lowing very Ample conftrudtion will determine this:
Let AOB (fig. 10.) be an overfhot wheel, of which
AB is the upright diameter, and C is the centre.
Make CF the length of a pendulum, which will make
two vibrations during one turn of the wheel. Draw
FE to the elbow of any of the buckets. The water in
this bucket, inftead of having its furface horizontal, as
NO, wall have it in the direction n O perpendicular to
FE very nearly.
For the time of falling along half of FC is to that of
two vibrations of this pendulum, or to the time of a re¬
volution of the wheel, as the radius of a circle is to its
circumference: and it is well known, that the time of
moving along half of AC, by the uniform action of the
centrifugal force, is to that of a revolution as the radius
of a circle to its circumference. Therefore the time of
defcribing one half of AC by the centrifugal force, is
equal to the time of defcribing one half of FC by gra¬
vity. Thefe fpaces, being fimilarly defcribed in equal
times, are proportional to the accelerating forces.
Therefore 4- FC : 4- AC, or FC : AC rr gravity : cen¬
trifugal force. Complete the parallelogram FCEK.
A particle at E is urged by its weight in the diredlion
KE, with a force which may be exprefied by FC or
KE; and it is urged by the centrifugal force in the
diredtion CE, with a force rr AC or CE. By their
combined adlion it is urged in the diredlion FE.
Therefore, as the furface of (landing water is always at
right angles to the adlion of gravity, that is, to the
plumb-line, fo the furface of the water in the revolving
bucket is perpendicular to the adlion of the combined
force FE.
Let NEO be the pofition of the bucket, which juft
holds all the water which it received as it paffed the
fpout when not affedled by the centrifugal force j and
let NDO be its pofition when it would be empty. Let
the vertical lines through D and E cut the circle de¬
fcribed round C with the radius CF in the points H
and I. Draw HC, IC, cutting the circle AOB in L
and M. Make the arch d’ 3 equal to AL, and the
arch e’s equal to AM : Then C $ and C e will be the
pofitions of the bucket on the revolving wheel, corre-
fpondirig to CDO and CEO on the wheel at reft. Wa¬
ter will begin to run out at 1, and it will be all gone at
—The demonftration is evident.
The force which now urges the wheel is (fill the
weight really in the buckets: For though the water be
urged in the diredlion with the force FE, one of its
conftituents, CE, has no tendency to impel the wheel 5
and KE is the only impelling force.
It is but of late years that mills have been conftrucled
or attended to with that accuracy and fcientific fkill
which are neceffary for deducing confidential conclufions
from any experiments that can be made with them ; and
it is therefore no matter of wonder that the opinions of
mill-wrightshavebeen fo different onthisfubjedl. There
is a natural wi(h to fee £ machine moving brifkly ; it has
the appearance of activity : but a very flow motion al¬
ways looks as if the machine were overloaded. For this
reafon mill-wrights have always yielded flowly, and
with fotne reludance, to the repeated advices of the
W AT
but they have yielded ; and we fee
them adopting maxims of conftrudion more agreeable to
found theory ; making their wheels of great breadth,
and loading them with a great deal of work. Mr Euler
fays, that the performance of the beft mill cannot ex¬
ceed that of the worft above fth : but we have feen a
Bream of water completely expended in driving a fmaU
flax mill, which now drives a cotton mill of 4000
fpindles, with all its carding, roving, and drawing
machinery, befides the lathes and other engines of the
fmith and carpenters workfhops, exerting a force not
lefs than ten times what fufficed for the flax mill.
The above difcuftion only demoni'trates in general the
advantage of flow motion ; but does not point out in
any degree the relation between the r£te of motion and
the work performed, nor even the principles on which
it depends. Yet this is a fubjeef fit for a mathematical
inveftigation ; and we would profecute it in this place,
if it were neceffary for the improvement of pradtical
mechanics. But we have feen that there is not, in the
nature of things, a maximum of performance attached
to any particular rate of motion which fliould therefore
be preferred. For this reafon v’e omit this difeuflion of
mere fpeculative curiofity. It is very intricate : For
■we muft not now exprefs the preflure on the wheel by a
conftant pillar of water incumbent on the extremity of
the horizontal arm, as we did before w hen we fuppofed
the buckets completely filled ; nor by a fmaller conjlant
pillar, correfponding to a fmaller but equal quantity ly¬
ing in every bucket. Each different velocity puts a
different quantity of water into the bucket as it pafie.s
the fpout ; and this occafions a difference in the place
where the difeharge is begun and completed. This cir-
cumftance is fome obftacle to the advantages of very
floAV motions, becaufe it brings on the difeharge fooner.
All-this may indeed be expreffed by a fimple equation
of eafy management ; but the whole procefs of the me¬
chanical difeuflion is both intricate and tedious, and the
refults are fo much diverfified by the forms of the buc¬
kets, that they do not afford any rule of fufficient gene¬
rality to reward our trouble. The curious reader may
fee a very full inveftigation of this fubjeiff in two differ-
tations by Elvius in the Swedifh Tranfafiiojns, and in
the Hydrodynamique o{ Profeflbr Karftner of Gottingen j
Avho has abridged thefe Diflertations of Elvius, and
confiderably improved the whole inveftigation, and has
added fome comparifons of his deduftions with the aftual
performance of fome great works. Thefe comparifons,
however, are not very fatisfadlory. There is alfo a
valuable paper on this fubjeft by Mr Lambert, in the
Memoirs of the Academy of Berlin for the year 1775.
From thefe differtations, and from the Hydrodynamique
of the abbe Boffut, the reader will get all that theory
can teach of the relation between the preffures of the
power and wbrk on the machine .and the rates of its
motion. The praftical reader may reft w’ith confidence
on the fimple demonftration we have given, that the
performance is improved by diminilhing the velocity.
All we have to do, therefore, is to load the macliine,
and thus to diminifti its fpeed, unlefs other phyfical cir-
cumftances throw obftacles in the way : but there are
fueh obftacles.^ In all machines there are little inequali¬
ties of a£lion that are unavoidable. In the a&ion of a
wheel and pinion, though made with theutmoft judge¬
ment and care, there are fuch inequalities. Thefe in-
creafs
Water¬
works.