Ae Quinby on the Overshot Water-Wheel. 307 
spate § is sufficient to raise the equal particle W through 
De vertical space, the thing proposed will be demon- 
si 
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” Pp"; which is equal to the 
sine sP’”; and, therefore, the effect of P, in descending 
through the element P” Pur, will be the same as_ the effect 
of P, in descending through the perpendicular 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. 
or. 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 ere depends (in 
theory) upon the fumber of buckets.”’ s error is con- 
spicucus in Dr. Gregory’s valuable anak on medics, and 
in the Edinburgh Encyclopedia. 
Next, let the wheel not be the whole height of the fall :— 
the same result will obtain. Suppose a particle of water 
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 es in giving angular motion to the 
wheel; or, in raising the equal particle W, Peepgpded 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 ~ acquired a velocity sufficient, if its direc- 
tion were reversed, to pr +r it back through the altitude 
' AF, to its pee height Whence it is manifest that the 
momentum of the particle P, in the direction AJ, will be 
: sufficient to raise the equal particle W, suspended at E, 
| through a vertical height= AF. Whereiore, as the effect af 
af ames 
* The author is aware that this part os this demonstration is not strietly 
: sedentitc. The result, however, is tru F 
