J. B. Quinby on the Overshot Water-Wheel, 307 



element, is sufficient to raise the equal particle W through 

 an equal vertical space, the thing proposed will be demon- 

 strated 



Let us take the element P'" P"" : and, then, since this 

 element is indefinitely small, we may consider it as coin- 

 ciding with the perpendicular P' P"" ; which is equal to the 

 sine sP'"; and, therefore, the effect of P, in descending 

 through the element P'" P"", will be the same as the effect 

 of P, in descending through the perpendi<;ular P P"" ;* 

 but, if P descend through the perpendicular P'" P"", it is 

 obvious that its effect (during the time of its descent) will be 

 sufficient to raise the equal particle W through the same ver- 

 tical space. Hence, by this, as by the preceding demonstra- 

 tion, any quantity of water, acting through any fall upon an 

 overshot water-wheel, which is the whole height of the fall, 

 will raise an equal quantity of water through the same verti- 

 cal height. 



Cor. From the foregoing demonstrations we are able to 

 correct the general error given by so many writers on the 

 overshot water-wheel, that " The effect produced by a given 

 quantity of water acting upon an overshot wheel, depends (iu 

 theory) upon the number of buckets." This error is con- 

 spicuous in Dr. Gregory's valuable work on mechanics, and 

 in the Edinburgh Encyclopsedia. 



Next, let the w-heel not be the whole height of the fall : — 

 the same result will obtain. Suppose a particle of water P, 

 to descend from F, Fig. 1, through the vertical height FA, 

 and to act, at A, upon the wheel, in the tangential direction 

 Al : — let us consider the effect that this particle will have 

 from its momentum at A ; or, which is the same, (P being a 

 unit,) from its velocity at A, in giving angular motion to the 

 wheel; or, in raising the equal particle W, suspended at the 

 extremity E, of the horizontal radius. By the known law of 

 falling bodies, the particle P, when it shall have arrived at the 

 point A, will have acquired a velocity sufficient, if its direc- 

 tion were reversed, to project it back through the altitude 

 AF, to its original height F. Whence it is manifest that the 

 momentum of the particle P, in^the direction A/, will be 

 sufficient to raise the equal particle W, suspended at E, 

 through a vertical height=AF. Wherefore, as the effect of 



* The author is aware that this part of this demonstration is not strictly 

 scientific. The result, however, is true. 



