308 A. BL Quaindby on the Overshot Water-Wheei. 
are tind is equal to that of the particle P, estimated for the 
same tim 
And as ibe effects produced by the two equal particles, 
(or powers,) P and P’, during any given time, will obviously 
be to each other, as the mean tendencies of those particles, 
(during the same time,) to produce rotation ; it follows that 
the effect produced by the particle P', in descending from A 
to B, in the are ADB, will be equal to that produced by the 
particle P, in the same time; or = WX Py. 
Hence, if a particle of water descend u upon an overshot 
_water-wheel, which is the whole height of the fall, it will 
= an equal particle through the same vertical height :— 
‘ and, as this will be the case with every particle, it ene — 
any quantity whatever of water, descending u 
hot water-wheel, which is the whole height of es “fall, will 
raise an equal quantity through the same vertical height. 
The same can be demonstrated in a different manner. 
Let the circle ABE, Fig. 2, represent an overshot wheel ; 
and let P, P’, P”, &c &c. be different situations of the same par- 
ticle of water P: and. suppose each of the arcs BP, 
&e. to be Gia tatiehaitely small element :—then, each element 
and its chord, (and also its sine and tangent.) may be con- 
sidered as coinciding. Wherefore, for the value of P, in the 
respective elements, we have, (by mechanics,) 
Px/im PXmn PXno | 
PP 7 Pp’ Pp” “pr py? Pp" a7 pm? pp” py ? 
and, for the effect of P in the respective elements, (or, for the 
tendency which P has in the respective star to raise the 
I PP’, 
Ul 
equal particle W,) we have pp oe al P or 
Pxmn a P Xno 
pr pe" Pp” Pp“ : pp” Pp’ eae paste Px<Pi, P xin, 
Pxmn, PXno; whence it is ae that the effect of P, in the 
respective elements s, is, always, as the perpendicular space 
passed through ; and, also, that 9 effect of P, in descending 
through any number of elements ; or through any portion 
whatever of the are P, P’, P’, be. is, always, as the pera 
