- 
304 A. B. Quinby on the Overshot Water-Wheel. 
nn—1. n—2 nn-l n—2n-3 
(A); iy (eye ee ES eat Ey etd be. 
=xX(A); it) stGy) Settee algae + Se. 
ro ot yt et Be ga (OE) pa to(A') 
and by: placing the factor n+1, at the beginning instead of at 
the end of the terms it becomes exactly like the formula (A’) 
whence the proportion assumed by Euler for integer posi- 
tive values of the exponent is true. 
Art. XIV.—New demonstrations on the theory of the Over- 
shot Rg hed By Mr. A. B. Quins 
Theo read? Any quantity of water, acting through any 
fall, upon an overshot water-wheel, will raise an equal quan- 
tity water through the same vertical height. 
t the wheel be the whole height of the fall: and 
destvitie the circle ADBE, Fig. 1, to represent the wheel. 
Draw the vertical diameter AB; 4 and at right angles to it, the 
diameter ay, now, as the quadrantal ar¢ “AD: sea 
CD : to a fourth term. Make CG=this fourth term; and 
suppose a wheel Gtvw, whose radius is equal to CG, ‘to be 
fitted permanently (in any way) upon the shaft that carries 
the water-wheel. Suppose, also, two racks, Gb and wd, to 
rest upon the teeth of the wheel Givw, and to stand parallel 
with the vertical diameter AB. If, now, a particle of water P, 
be applied upon the end of the rack Gd, it is obvious that it 
will cause this rack to descend, and turn the wheel Gtvw, and 
“nes we rack vd on the opposite side; and if a particle of 
=to P, be attached to the lower end of the rack vd, 
, is 4 plas that the two particles, P and W, will reciprocally 
balance each other ; and, if the particle P be supposed to de- 
scend through any space whatever, its effect, during the time 
* For this description see Plates TV. and V, 
