Or WATER MILLS, &e. 147 
judged by the fociety, and prefented'to the author by their 
prefident. An inftance ‘or two of the Importance of this 
correction may be adduced as follow. 
Prop. II. 
The velocity of a wheel, moved by the impaét of a 
fiream, muft be half the velocity of the fluid, to produce 
the greaieit poflible effed. e 
Fo: le V=the velocity, M—the momentum of the fluid 
v= the velocity, P =the power of the wheel. 
Then, V—v = their relative velocity, by definition 3d. 
and, as Vi: V—v:: M:¥ x Voy=P (Prop. 1.) which 
x v= Pv i272 x vvev2_ a maximum; hence v v-v2= a 
maximum, and ‘its fluxion, (v being the variable quantity) 
=-Vv —2vy =0; therefore v= 4, that is, the velocity 
of the wheel — half that of the fluid, at the place of im- 
pact, when the effe&t is a maximum. 0. Ea De 
_The ufual theory gives v= V; where the error is not 
lefs than one third of the true velocity of the wheel! 
This propofition is applicable to underfhot wheels, and 
correfponds with the accurate experiments before cited, as 
appears from the Author’s conclufion, (Philofophical 
TranfaQions for 1776 pase 457) viz. “ The velocity of 
‘“‘ the wheel, which, according to M. Parents determina-~ 
‘* tion, adopted by Defaguliers and Maclaurin, ought to 
** be no more than one third of that of the water, varies 
**at the maximum jn the experiments of Table I. be- 
‘** tween one third and one half; but in all the cafes there 
“* related, in whichthe moft work is performed in propor- 
“ tion to the water expended and which approach the near- 
eft to the circumftances of great works when properly 
“ executed, the maximum lies much nearer one half than 
“ one third, one half feeming to be the true maximum, if 
** nothing were loft by the refiftance of the air, the fcatte- 
*¢ ring of the water carricd up by the wheel, &c.”? Thus 
eg he 
