TOL. LVII.J PHILOSOPHICAL TRANSACTIONS. 451 



when the first is removed from the vertical as far as a certain point, the next 

 may occupy that advantageous place, to be in its turn replaced some time after 

 by a third, and so on. Now our third inquiry is, to assign the angle contained 

 between two float-boards, or, which comes to the same, the number of float- 

 boards the wheel should consist of, that its efi^ct may be the greatest possible; 

 being of no less importance than the preceding ones. To begin then with the 

 most simple case; we will suppose the wheel immoveable, or that c = O, and 

 proceed to investigate, whether, supposing the number of float-boards to be 

 greater, the sum of the effects will come out greater or less than what results 

 from one single float-board placed vertically. 



In order to a general solution of this question, we will suppose 2 float-boards 

 CD and CE, fig. 11, making any angles with the vertical; and let us compare the 

 effect of the single float-board gd, with the effect resulting from the float-boards 

 PE and GD taken together, which will be reduced to pe and od, because the 

 part OG becomes useless, as the stream is intercepted by pe. Let cb = cd = 



CB = a, cti. = f, cosin. bcd = m, cosin. bce = (/., which gives cg = 



f f 



'i- cp = - and CO = ma. Then we shall find, by § vi, the effect of gd = t-^-j- 



nlrw (mmaa — Jf), that of od = -r^nbvv (mmaa — i^!*aa), and that of pe = 

 ■j-^nbvv (fxixaa — Jf); whence it appears, that the sum of the last two is 

 exactly equal to the first, which will ever hold good whatever be the value of J". 

 Whence arises the following theorem : " Whether the wheel be plunged quite 

 up to the axle, or only in part so, provided it be immoveable, and that one of 

 its float-boards be placed vertically, its effects will be constantly the same, what- 

 ever be the number of float-boards opposed to the stream, even though it were 

 infinite." The latter part of this theorem, though flowing from the general 

 demonstration, may be also demonstrated, immediately, in the following manner: 



let BP, fig. 12, be = X, we have 



adx J aa — dx — ax ., a* — Za?x + a'x' — Za'dx + Zaaxdx 



MO = , and CO = ; co^ = ■- , neer- 



a — X a — I aa— 2ax + xx ' & 



lecting the dx'^, and aa — co'* = ; therefore the effect of the stream on 



° ' a — V 



CM, which is = -r-h,- nbvv {aa — co*") — -^, will become = -^-^nbvv(2ax—2xdx), 

 whose integral is = -^^-5- nbvv {2ax — xx), where putting x = a —f, we have 

 -pi-o '"■^'^'^ (fl^ — ff) for the total effi?ct of the stream on the wheel ; which is the 

 same as that of a single float-board ab in a vertical position. 



^ VIII. This theorem will also hold true for the case of ^ v, where we have 

 supposed the height of the float-boards very small, in comparison of the radius 

 of the wheel ; we have seen that the effect of a single float-board placed verti- 

 cally was = nabvv (2r -|- a) ; the demonstration of the preceding § will be appli- 

 cable here after the same manner, and will show that whatever be the number of 



3m 2 



