VOL. LVn.] PHILOSOPHICAL TRANSACTIONS. 447 



other, and give motion to the part destined to produce some purposed effect, as 

 the mill-stone in a corn-mill. The size of the float-boards, the velocity with 

 which the wheel is to turn, and the number of float-boards to produce the great- 

 est possible effect, are 3 main things proposed to be examined in the following 

 inquiry. In the first place, Mr. M. supposes the total resistance which this 

 wheel has to encounter, on the part of the machine, and which hinders it from 

 moving so swift as the stream, to be expressed by a weight tt, suspended to the 

 extremity of a cord fixed to the circumference of a wheel wliose radius isd, and 

 which has the same axle as the float-board wheel, so that the efl^ect of the stream 

 is to raise the said weight tt, as expressed in fig. 8, pi. 10. He also supposes, 

 that the stream, by its velocity, moves through v feet in one second of time, and 

 that this velocity is the same, though at different depths. 



§ II. After these suppositions, the first thing that presents, is to determine 

 what should be the size of the float-boards, for the stream to be capable of raising 

 the weight v with a certain determinate velocity. Let aa bb, fig. Q, be one of 

 the float-boards let into the axle aa, and placed vertically in the water, so as to 

 receive the perpendicular impulse of the stream. Its horizontal length bb = b 

 feet, its vertical height ar = a feet, the velocity of the wheel at the point b, 

 such that it shall run through z feet in a second; n pounds the weight of a cubic 

 foot of water ; and suppose the impulse of the stream on a plane perpendicular 

 to it is (as Dr. Daniel Bernoulli has stated in his Hydrodynamica) equal to the 

 weight of a prism of water, whose base is the plane, and its altitude the gene- 

 rating height of the velocity with which the plane is impelled. This being sup- 

 posed; let AP ^ X, vp, its differential, = dx, which will give the velocity of the 



float-board at the point p =i -z, and the relative velocity of the stream with 



which the plane is impelled at the same point = v 



z, whose generating height is ^ (v z)* feet ; whence we have the weight 



of the parallelepiped of that height, and of the base pp pp equal to -g- (v - z)', 



pounds, which weight multiplied by the length ap (x) of the lever which tends 



to turn the plane, gives —. — {v z)* for the total effect of the stream on the 



little rectangle vr pp, whose integral is 



^ (iwxa; - -|-t;z- 4- izz -^ - \vvfj -^ \vz. -'- - i -^), puttmg ac =/ 

 for the distance between the axle and the surface of the water when the float- 

 board has only its part cb plunged in the water ; which, putting x ■=. a, becomes 

 ~ {\vv— \vz -f- 4-zz. aa — \vvff -\- «- '^ — 4. ??^), which will express the 



effect on the whole plane ccbb, equal nd, the product of the weight ■k by the 

 length d of the lever on which it acts in opposing the motion of the wheel. 



