308 ./f. 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 WxBA; we have, for the whole effect of the parti- 

 cle P, during its descent from F to B, W X AF+W xBA= 

 WxBF:* — and as this result will obtain for any value what- 

 ever of AF, it follows, that if a particle of water descend 

 through any fi\ll, and act upon an overshot water-wheei, 

 which is northe whole height of the 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 water, acting through any fall, upon an 

 overshot water-wheel which is not the whole height of the 

 fall, will raise an equal quantity 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 

 effect of any body in motion varies as the square of its mo- 

 mentum Wherefore, putting A = FB, Fig 1, and a;=FA; 

 and supposing a particle of water P, to descend from F to A, 

 we shall have, for its effect at A, (estimated in the direction 



A?,) (P\/2ga;)2 ; and for its effect in descending from A, 



through ADB, we shall have (P^/'2gXh — x) '^. Hence, 

 for the whole effect of P, in descending from F to B, we have 



{F^^gxy+{P'K/2gy<h-xy=F'-2gx+F'-2gXh-x=2P^- 

 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, consequently, the same result ob- 

 tains in the second case as in the first. 



There is still another method by which this second case 

 can be demonstrated. 



Let am6E, Fig. 3, represent an overshot wheel ; and AF 

 the height of the fall above the wheel : and suppose a tight 

 canal, or conduit, Fambic, to be filled with water, and to be 

 supplied from the trough GF ; then, by a known law in hy- 

 draulics, any quantity of water which shall descend from F, 

 will cause an equal quantity to be discharged at c; and with 

 a velocity that will project it through the vertical space cd, 

 to the original summit level FGd. 



* In this, and in some other of the demonstrations here given, the 

 wheel js suppose d to move indefinitely slow. 



