308 =. ol. B. Quinby on the Overshot Water-Wheel. 
P, after it shall have issued upon the wheel at A, and during 
its descent in the arc ADB, will, by the foregoing case, be 
equal to WX BA; we have, for the whole effect of the parti- 
cle P, during its descent from F to B, WX AF+ B 
W x BF: :*—and as this result will a forany value what- 
ever of AF, it- Sollee wl - a he of water descend 
through any fall, and a an overshot water-wheei, 
which is not the whole Son of ie fall, it will raise an equal 
particle through the same vertical height. And, as the same 
will be the case for any number of particles, it is demonstrated 
that any quantity of ad acting through any fa all, upon an 
overshot water-wheel which is not the whole height of the 
a ll, will raise an equal aaah of water through the same 
vertical height. 
There is another method by which this second case can 
be demonstrated by referring to the first. By mechanics the 
_ of eae body in motion varies as the square of its mo- 
men . Wherefore, putting A=FB, Fig 1, and c=FA; 
and continiiiee particle of water P, to descend from F to A, 
we shall have, for its effect at A, (estimated in the direction 
Al.) (Py 3g) #; and for its effect chet from A, 
through A ADB, we shall have (P./2¢ Xh—a@)?. Hence, 
for the whole effect of P, in in descending frome F to B, we have 
(Px/2¢2)? +(P/2¢ xh— =a)? =P22¢n4P29¢ Xxh— -2==9P? 
gh—a constant quantity; and which shows that the effect of 
an overshot water-wheel does not depend, at all, upon the 
wheel’s diameter :—and, reheat hint , the same “palate ob- 
tains in the second case as in the 
There is still another method by ‘which this second case 
can ri demonstrated. 
t ambE, Fig. 3, represent an overshot wheel; and AF 
the height of the fall above the wheel: and suppose a tight 
canal, or conduit, Fambtc, to be filled with water, and to be 
supplied from the trough GF; then, by a known law in hy- 
draulies, any quantity of water which shall descend from I, 
will cause an equal quantity to be discharged at ¢; and with 
a velocity that will project it through a vertical space ed, 
to the original sammit level FGd. 
_ *In this, and in some other of the demonstrations here given, the 
_ wheel is supposed to move indefinitely slows, 
