432 PHILOSOPHICAL TRANSACTIONS. [aNMO 1767^ 



float-boards, the effect will be ever the same. It does not however follow that 

 the number of float-boards should be indifferent; for the wheel coming to turn 

 the float-board, its lower part, which received the perpendicular impulse, will no 

 longer receive it otherwise than obliquely, and the eftect will diminish till the 

 angle formed by two neighbouring float-boards be bisected exactly by the vertical, 

 v/hich will render the first entirely useless; after which the eftect will increase 

 anew, and will become again greatest when the 2d float-board is got to the ver- 

 tical ; so that in order to fix on the most advantageous number of float-boards, 

 regard must be had to the sum of the dififerent effects for all the situations of the 

 float- boards during one whole turn of the wheel. Whence it follows, that in 

 this case, wherein they are supposed very small, the greater their number is, the 

 greater will be the sum total of the effects ; since, if that number were infinite^ 

 there would be a float-board in a vertical position every instant. 



^ IX. This will no longer hold good, if the height of the float-boards be more 

 considerable, arid it be found necessary to take the different velocity of their dif- 

 ferent jx)ints into consideration; by comparing, fig. 11, the pressure on fe with 

 that on the portion go, they will be found no longer equal, as in the foregoing 

 case; it is true that the same quantity of fluid acts on these two planes, and the 

 disadvantage which fe has by receiving the impulse more obliquely, is exactly 

 compensated, as before, by the length of the lever, but the difference arises from 

 the different velocity of the corresponding points of fe and go ; those velocities 

 are in the ratio of cf to cg, or as cos. acg to cos. acf, which shows that the 

 effect of FE is always less than that of go, and consequently the effect must be 

 diminished, by adding a greater number of float-boards: the said eftect will be 

 greatest when there is only one float-board placed vertically, and least when their 

 number is infinite: let us inquire what it will be in this latter case. We will 

 suppose the same, fig. 12, and the same denominations as in ^ vii. We had 

 CO = °" ~ °^ ~ " -, we shall have, (neglecting cbr', dx^, and dx*), aa — co" = 



tzfUL and = . Now the pressure on om is, by is ii, = 



a — i' a ' aa a — x ' ^ j :i ' 



-ri^ nh ^" ~j'^\ \yv {aa — co") — ^ vz. ° ~ ""^ + 4- zz. a* — ^] 5 which, (by 

 putting for co its value) will become = -f^ nb {2adx — 2xdx) . {v — z'^), whose 

 integral -rH- ^^ {'^"^ — ^^) ■ (^ — ^^) = t-to nb (v — z') {aa — Jf), making 

 X = a — /, will express the effect resulting from an infinite number of float- 

 boards: this least effect will be to the greatest, that is when there is but one 

 float-board, as {v — z'^) : vv — 



- — . — ^. -\- 4r ~. {aa +JY), or as {v — x^) : -^vv — i zz {—-'■), ^ vi. 

 3 a aa — ff ' ^ aa ^ i^^//' \ /« ■» ^ aa ' ' 



This ratio will be that of 1 : 2 if/ = 



1 : 1 .485 ^ / = io 



1:1 " / = «• 



