156 On the Measure of 
- Mr. Waring’s demonstration is: as follows : 
“Tf the relative velocity of a fluid against a 
siigle plane be’ variéd, either by the motion of 
the plane, or of the fluid from a given aperture, 
or both, then, the number of particles acting 
on the plane in a given time, and likewise the 
momentum of each particle, being respectively 
as the relative velocity, the force on. both these 
accounts, must be-in the duplicate ratio of the 
relative velocity, agreeably to the common 
theory, with respect to this single plane ; but, 
the number of these planes, or parts of the 
wheel acted on in a given time, will be as the 
velocity of the, wheel, or inversely as the rela- 
tive velocity; therefore the moving force of 
the wheel must be in the simple direct ratio of 
the relative velocity,” and, consequently, the 
maximum effect must be produced when the 
velocity of the wheel is half that of the 
water.* 
. But this kind. of demonstration cannot, I 
think, be very satisfactory. It leads, I appre- 
hend, to this conclusion, that we may double 
the power of any undershot water-wheel, 
(whatever may be its velocity) by merely 
doubling the number of its floats or planes 
acted upon by the w ater. Mr. Smeaton, how- 
ae 
* Ameriezn Philos, Trans. vol. S$. p. 146, 
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