450 PHILOSOPHICAL TRANSACTIONS. [aNITO I767. 



the velocity — — z. The exact solution of this problem might be deduced from 

 the formula, § ii, which would render the operation tedious, the equation being 

 of the 4th degree; but it may be rendered much simpler by a supposition which 

 is but little wide of the truth when ab is but small in comparison of ca; and 

 this is, to consider all the points of the float-board ab as affected with the same 

 velocity z. 



Let cp, fig. \0, z=z X, we shall have -^nb^v — zf Jl.xdx for the effect of the 

 portion ap, and -^nh (v -— z)' X {ar + ad) for the effect of the whole float- 

 board AB. This quantity must be made equal to rfir, and then, just as in the 

 foregoing cases, such a value of z be sought, that the weight rr and its velocity 

 may be the greatest possible; that is, the differential of z (w — zf must be made 



= 0, which gives z = \v. Therefore d-a = - J (2r -|- a), and r 



= — - — 7 J and the ve ocity of the weight t will be = — — ; ; — x v. 



§ VI. We have seen that the calculus was much simplified by supposing one 

 of the velocities constant for all points of the float-board. For this velocity being 

 c, the effect of the whole float-board will be simply ■g'^nb (vcy (aa — -k^)- It 

 will therefore not be unuseful to inquire what this velocity c must be, that the 

 effect of the float-board may be the same, as supposing, as we have hitherto done, 

 a variable velocity, and proportional to the distances from the axle, we have 

 only to make 

 _ (y _ e)^ (— ^) = ^ i±vv {aa -ff) - ^vz —^ -\- \zz —^), § iv ; 



by the equation whence we got the value of z § iv, \vv — - - . °, _ ■ -|- ^zz. 



— — — = 0, we shall have c = v \/-irVV — 4-zz. ^^^^ — —. 



aa ' ^ ^ " aa 



If/= 0, we have zz = 0.288iw and c = 0.345t;, 



f=-^a zz = O.igOw c = O.330i;, 



f= a zz = -^vv c = -^v, 



so that whatever be the relation of /"to a, the velocity c is ever nearly = Iv, and 

 the more exactly so, as /is greater. Therefore we may always assume ,4o nbtjv. 

 (aa — ff) = dir for the effect of the stream on a float-board whose plunged part is 

 a —f; this effect will be increased in the ratio of 4 to Q, when the wheel has no 

 motion, for making c = O, we find it = T-fj-. nbvv. {aa — ff). 



§ VII. Hitherto we have all along supposed that the float-board, through its 

 whole plunged part, received the perpendicular impulse of the stream; but it is 

 easily understood that the wheel, coming to turn, presents to the stream the 

 plane of the float-board under an angle which is continually varying, which 

 diminishes its effect every instant as it removes from the vertical : this inconve^ 

 nience may be remedied by multiplying the number of the float-boards, so that 



