M E C H A N I C S. 
7 04 
rent. The wheel, therefore, mud move irregularly, fome- 
tiines quick and fometimes flow, according to the pofition 
of the floatboards with refpeft to the ftream; and this in¬ 
equality will inoreafe with the arc plunged in the water. 
The reasoning of M. Pitot, indeed, ij founded on the 
fuppofition, that, if another floatboard fg were placed 
between FG and D F, it would annihilate the force of 
the water that impels it, and prevent any of the fluid from 
finking the correfponding part DO of the preceding float- 
board. But this is not the cafe ; for, when the water has 
a£led upon fg, it (fill retains a part of its motion, and, 
after bending round the extremity g, ftrikes DE with its 
remaining force. We are entitled, therefore, to conclude, 
that advantage mu ft be gained by ufing more floatboards 
than are recommended by Pitot. 
It is evident from the preceding remarks, that, in or¬ 
der to remove any inequality of motion in the wheel, and 
prevent the water from efcaping below the extremities of 
the floatboards, the wheel fhould be furnifhed with the 
greateft poflible number of floatboards, without loading it 
too much, or enfeebling the rim on which they are fixed. 
This rule was firlt given by M. Dupetit Vandin, in the 
Mem. des Sfavans Etrangers, tom. i. and it is eafily per¬ 
ceived, that, if the millwright fhould err in ufing too 
many floatboards, this error in excefs will be trifling, and 
that a much greater lofs of power would be occauoned by 
an error in defeat. 
Thefection of the floatboards ought not to be reftan- 
gular like abnc, but bevelled like abmc-, for, if they were 
rectangular, the extremity bn would interrupt a portion 
of the water which would otherwife fall on the corre¬ 
fponding part of the preceding floatboard. In order to 
find the angle abm, fubtraCl from 80 degrees the number 
of degrees contained in the immerled arc C E G, and the 
half of the remainder will be the angle required. 
It has been maintained by M. Pitot and other philofo- 
phers, that the floatboards (hould be a continuation of 
the radius, or perpendicular to the rim, as in fig. 99. This 
indeed is true in theory; but it appears from the molt 
unqueftionable experiments, that they fhould be inclined 
to the radius. This important faft was difeovered by 
Deparcieux in 1753, and proved by feveral experiments. 
When the floatboards are inclined, the water heaps up on 
their furface, and afts not only by its impulfe, but alfo 
by its weight. The fame truth has alfo been confirmed 
by the abbe Boffut, the molt accurate of whofe experi¬ 
ments are contained in the following Table. The wheel 
that was employed was immerfed four inches vertically in 
the water, and it was furnifhed with twelve floatboards. 
Inclination 
of the 
Floatboard. 
Number 
of Pounds 
raifed. 
Time in which 
the Load was 
raifed in feconds 
Number of 
Turns made by 
the Wheel. 
O 
4-0 
40 
15 
40 
40 
30 
40 
40 
hH 
; 37 
40 
40 
I 
2 
3 
4- ! 
It is obvious, from the preceding Table, that the wheel 
made the greattfl number of turns, or moved with the 
greateft velocity, when the number of floatboards was be¬ 
tween 15 and 30. When the water-wheels are placed on 
-canals that have little declivity, and in which the water 
-can efcape freely after its impulfe upon the floatboards, it 
would be proper to make the floatboards a continuation 
of the radius; but, when they move in an inclined mill- 
courfe, an augmentation of velocity may be expetted from 
an inclination of the floatboards. 
Having thus pointed out the moft fcientific method of 
conftrufting the wheel, and delivering the water upon it's 
.floatboards, we have now to determine the velocity with 
which it fhould move. It is evident, that the velocity of 
the wheel muft be always lefs than that of the water which 
impels it, even when there is no work to be performed; 
for a part of the impelling power is neceflarily fpent in 
overcoming the inertia of the wheel and the reflftance of 
friction. It is likewife obvious, that, when the wheel has 
little or no velocity, its performance will be very trifling. 
There is, confequently, a certain proportion between the 
velocity of the water and the wheel, when its effect is a 
maximum. Parent and Pitot found, that a maximum 
efreCl was produced when the velocity of the wheel was 
one-third of the velocity of the water; and Defaguliers, 
Maclaurin, Lambert, and Atwood, have adopted their 
conclufions. In the calculus from which this refult was 
deduced, it was taken for granted, that the momentum or 
force of water upon the wheel is in the duplicate ratio of 
the relative velocity, or as the fquare of the difference 
between the velocity of the water and that of the wheel. 
This fuppofition, indeed, is perfectly correCt when the 
water impels a (ingle floatboard; for, as the number of 
particles which (trike the floatboard in a given time, and 
alfo the momentum of thefe, are each as the relative velo¬ 
city of the floatboards, the momentum muft be as the 
fquare of the relative velocity, that is, M==R 2 , M being 
the momentum and R the relative velocity. But we 
have feen, in fome of the preceding paragraphs, that the 
water aCts on more than one floatboard at a time. Now 
the number of floatboards acted upon in a given time will 
be as the velocity of the wheel, or inverfely as the rela¬ 
tive velocity j for, if you increafe the relative velocity, 
the velocity of the water remaining the fame, you muft 
diminifh the velocity of the wheel. Confequently, we 
R2 
(hall have M== — or M==R; that is, the momentum of 
■ R 
the water acting upon the wheel, is direftly as the rela¬ 
tive velocity. 
Let V be now the velocity of the ftream, and F the 
force with which it would (trike the floatboard at reft, 
and v the velocity of the wheel. Then the relative velo¬ 
city will be V— v ; and, (ince the veiocity of the water will 
be to its momentum, or the force which it would (hike the 
floatboard at reft, as the relative velocity is to the real force 
which the water exerts againft the moving floatboards, 
„ „ , V — v F 
we mail have V : V—1/ = F : Fx~ y- =~XV— v. But 
the effeCt of the wheel is meafured by the product of the 
momentum of the water and the velocity of the wheel 3 
confequently the effect of the underftiot-wheel will be 
F F 
v X — X v — v —~y X Vw — Now ^ 11S effeil is to be 
a maximum, and therefore its fluxion muft be equal to o; 
that is, v being the variable quantity, Vv — 2vv — o; 
or 2W — Vv. Dividing by v, we have 2i/ = V, and 
V 
—, that is, the velocity of the wheel will be one half the 
velocity of the fluid when the effect is a maximum. 
An under(hot-wheel, with oblique floatboards, was in¬ 
vented by the late ingenious Mr. Befant, of Brompton ; 
on whofe widow the Society for the Encouragement of 
Arts, &c. in 1801, conferred a reward of ten guineas; 
and, as it may be of great fervice in many lituations, we 
have given a reprefentation of it from the xixth vol. of 
their TranfaCtions. A, fig. 102. reprefents the body of the 
water-wheel, which is hollow, in the form of a drum, and 
is fo conftrufted as to refill the admiftion of water. B is 
the axis on which the wheel turns. C C C, the float- 
boards, placed on the periphery of the wheel, each Gf 
which is firmly fixed to its rim and to the body-of the 
drum, in an oblique direction. D is the refervoir that 
contains the water. E, the penltock, for regulating the 
quantity of water which runs to the wheel. F repreients 
the current that has paffed fuch wheel. Fig. 103. is a 
front view of the water-wheel, exhibiting the oblique di¬ 
rection 
