304 A. B. Quinby on the Overshot Water-Wheel. 



(A); l+j-x+(j).— .^^4-(-j -^) -^ ^»+(i -^•^)t-^''+ ^^' 



1 n 2 nn-\^ 3 nn-\ n-2 4 



^X(A); -.^+(-) 2-^^+(v^) 3-^'+(r-F- 3~^4-^* + *''^- 



and by placing the factor n-\-\, 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 ofthe exponent is true. 



AiiT. XIV. — JVezo demonstrations on the theory of the Over- 

 shot Water-wheel. By Mr. A. B. Quinbt.* 



Theorem I. Any quantity of water, acting through any 

 fall, upon an overshot water-wheel, will raise an equal quan- 

 tity of water through the same vertical height. 



First, Let the wheel be the whole height of the fall : and 

 describe the circle ADBE, Pig, 1, to represent the wheel. 

 Draw the vertical diameter AB ; and at right angles to it, the 

 diameter ED. Say, now. as the quadrantal arc AD : CD;: 

 CD : to a fourth term. Make C 6 = this fourth term ; and 

 suppose a wheel Givw, 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. GZ> and vd, to 

 rest upon the teeth of the- wheel G>V7d, and to stand parallel 

 with the vertical diriraeter AB. If, now, a particle of water P, 

 be applied upon the end of the rack G5, it is obvious that it 

 will cause this rack to descend, and turn the wheel Gtviv^ and 

 raise the rack vd on the opposite side ; and if a particle of 

 water W=to P, be attached to the lower end of the rack vd, 

 it is plain 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 IV. and V. 



