MECHANICS.

61

Chap. 2.]

only on account of the varied impingent velocity of thesame body, occasioned by a change of motion in thewheel; that is, the momentum is as the relative velocity.

Now, this true principle, substituted for the erroneousone in use, will bring the theory to agree remarkablywith the notable experiments of the ingenious Smeaton,published in the Philosophical Transactions of the RoyalSociety of London, for the year 1751, vol. 51; forwhich the honorary annual medal was adjudged by thesociety, and presented to the author by their president.

An instance or two of the importance of this correctionmay be adduced, as follows:

Prop. ii.

The velocity of a wheel, moved by the impact of astream, must be half the velocity of the fluid, to pro-duce the greatest effect possible.

C 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 Y:Y—v::M:^xV—v=P, (Prop. 1,) which

V

Xv=P, v=m x y v —v J =a maximum; hence Vv—v 3 =

V

a maximum and its fluxion (v being a variable quantity)=Vv—2vv=0; therefore =JV; that is, the velocity ofthe wheel=half that of the fluid, at the place of impactwhen the effect is a maximum. Q. E. D.

The usual theory gives v=JV, where the error is notless than one-sixth of the true velocity.

Philadelphia , 7th9th mo. 1790.

Wm. Waring.”