YOL. LVn.J PHILOSOPHICAL TRANSACTIONS. 441) 



If/ = 0. this value of z = 0.537» 



/= 0, 25a = 4a = 0.498» 



/=0. 3a = ().486» 



/= 0.5a = Ja = 0.436t) 



/= 0.7a = O.39OD 



f=0.9a = 0.35311 



f=a = 0.3331- 



By the inspection of these different values it appears, that this value of z di- 

 minishes as the plunged part is greater, and that this velocity can never exceed 

 the quantity 0.537v, nor be less than ^v.* ,j 



•St This value of z^ and of its square zz, being substituted in the general formula, 

 § II, we shall obtain from it the following equation : 



60rfT_ 1 laf-Q7a^f^+ 32af'--27a^f+ 1 1/' + 2 (a'-a/Q ^(lOa'' -\-Sia*f* - 1 28a'/^ + 54ay + 10/^) 

 nbvv~ 81 (a* -/^) 



which, for a given relation between Z" and a. will show the breadth, b for pro- 



, . ' ° „ ; n r,,/«'ri~ii >!!^ -xii iiiif-! no ' •;,(;!; ■ ■/ ' ^ 



ducing the greatest etiect. , . " , . , . 



As the extremity of the float-board must have a certain velocity depending on 

 tlie relation of the height a to the plunged part, and as the velocity of the weight 

 w s= - z, it follows that if we should increase the velocity of tt, we must dimi- 

 nish the height a and increase the breadth b, so that the product baa and the 

 relation ofyto a may be the same as before. For example, if the wheel be 

 plunged as deep as the axle, to double the velocity of the weight, the height of 

 the float-board must be- reduced one half, and its breadth be quadrupled. 



§ V. It may so happen, that the channel on which the wheel is placed shall 

 be so shallow and narrow, as not to allow the float-boards the necessary dimen- 

 sions for raising the weight with a convenient velocity. In this case we are 

 obliged to raise the axle of the wheel alx)ve the surface of the water so much, 

 that the lever on which the stream acts may be long enough to recompense the 

 smallness of the float-boards. Herein it is necessary to solve the following pro- 

 blem. The breadth b and the height a, of the float-board ab being given; to 

 find the radius ca (r) of the wheel which shall cause the weight tt to ascend with 



* If in the value of z we make/= a, we have z = ^. which obliges us to take, according to the 



common method, the differentials of the numerator and of the denominator, considering/ as variable, 

 and the relation of these diiferentials, will give the value of z ; but on account of the radical quan- 

 tity, the calculus being somewhat tedious, and again bringing out* = jtj and that after several si- 

 milar operations, it is better to have recourse to the equation from which the value of z was deduced j 



a* ~ fi a' — P 



this equation is jaw. -; — ^ — Joaov. —nryi~> which by the above operation will be jiz = vx 



— \vB, and z =: \v. 

 VOL. XII. 3 M 



