HYDRODYNAMICS. 



557 



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The experiments of Bossut were made with a wheel, 

 whose exterior diameter WM 3 feet 1 inch 1 lines. It 

 WM used successively with 48, 24, and 12 floatboards 

 directed to the centre. They were exactly 5 inches 

 wide, and from 4 to 5 incites high. The edges and the 

 extremities of the floatboards- were distant about half a 

 line from the bottom and side* of the inclined canal in 

 which the wheel WM placed, and the arch plunged in 

 the water WM 24 54'. When this wheel WM tried, it 

 made 33$ turns in a minute, when it had 48 floatboards, 

 and when the weight raised was 12 pound*. When 24 

 floatboards were put on, it made only 29 turns in a mi- 

 nute, the weight raised being the same ; and when 12 

 floatboards were used, it made no more than 25^ turns 

 in a minute. The velocity of the water in the canal, 

 which had a declivity of 1O$ feet in 50, was 300 feet in 

 33 second*. Hence Bossut concludes, that the wheel 

 ought to have at least 48 floatboards, whereas wheels 

 feet in diameter, and with only 2.i or 30 of the 

 circumference immersed, have generally only 40. 



When wheels are moved by a river, they ought to 

 have a different number of floatboards. In order to find 

 the number, M. Bo-sut used a different wheel, in which 

 the floatboards were so placed that he could *et them 

 at any inclination to the radius, and employ any number 

 of them at pleasure. The exterior diameter WM 3 feet, 

 the width of the floatboards 5 inches, and their height 

 6 inches. This wheel WM made to move in a current 

 12 to 13 feet wide, and in a depth of water of 

 7 to inches. The floatboards were plunged four 

 in the water, so that the circumstance* were the 

 same M in an open river. When 2* floatboard* were 

 *ed, a load of 40 pound* WM raised with a velocity of 

 1 SIT turn* hi 40 econds ; whereM when 12 floatboard* 

 were used, the velocity with which the same load WM 

 raUed WM only 13)J turn* in the same time. When 

 48 flout board* were put on, 24 pounds were raised, with 

 u vssecitj of 27 J4 turn* in a minute ; and 24 floatboard* 

 raised the weight with a velocity of 27 rV- ** dJuVranoe 

 bean* perfectly triftmg. Hence 94 floatboard* at least 

 should b* used hi CMC* of this kind. A smaller num- 

 ber would be sufficient, if a greater arch of the wheel 

 i plunged in the stream. In practice, it WM the 

 to use only 8 or 10 floatbuurd*, and 

 r, on wheel* placed in rivers ; but the 

 - ought to have been from 14 to 



From theoretical consideration*, M. Pilot ha* shewn, 

 that floatboard* should always be a continuation of the 

 radiut ; but this rule ia found to be quite incorrect in 

 practice. The advantage* arising from inclining the 

 floatboard^ were first pointed out in 1 T.J.'-; by Drpar- 

 cieux, who shews, that the water will thus heap up upon 

 them, and act by its weight as well a* by it* iropuUe. 

 This opinion fvos been amply confirmed by the cxperi- 

 . menu of Bouot with the wheel already mentioned, mov- 

 ing ui a canal where the velocity of the water WM 300 

 feet in 7 seconds. When the floatowrd* were a con. 

 donation of the radius, a weight of 34 pounds WM rais- 

 ed with a velocity of *')i turn* in 40 second*. When 

 their inclination WM 8*, the same load WM raised with 

 a velocity of li$ in 40 second*. When tlie inclination 

 . : .'-, the velocity WM 194$ in 40* ; "d w hcn tlie 



imlinalion WM llj", the velocity WM 904 turn- in 40 se- 

 conds, nearly the same, but still a little less than when 

 the floatboerds were a continuation of the radiiu. Hence 

 w, that a wheel placed upon canals which have 

 little declivity, and in which the water is at liberty to 

 eacape easily after the impulse, the floatboards ought 

 10 be a continuation of tht radiut 



The same wheel bein<; placed in the current already 

 mentioned, vie. from 12 to 13 feet wide, and from 7 to 

 8 inches deep, floatboards which were a continuation 

 of the radius, raided 40 pourxls with a velocity of 13^ 

 turns in 40 seconds. With those inclined J 5, the num- 

 ber of turns in the same time was 144^ : with those in- 

 clined 30, the number was H|4 J an< ^ w ith those in- 

 clined 37, the number was 14J|. Hence it follows, 

 that the moft advantageous obliquity is, in this case, 

 about 1 5 or 30 degrees. The difference of effect, how- 

 ever, appears to be very trifling, particularly beyond 

 15. M. Fabre is of opinion, that when the velocity of 

 the stream is 1 1 feet per second or greater, the inclination 

 should never be less than 30" ; that, as the velocity di- 

 minishes, the number of floatboards should diminish in 

 proportion ; and that when the velocity is 4 feet or un- 

 der, the floatboards should be a continuation of the radius. 

 The experiment of inclining the floatboards a little in 

 the opposite direction, has not been tried by anjr of the 

 authors whom we have quoted, but we think it worth 

 trying, M it might increase the effect, by allowing the 

 water to escape more readily from below the float- 

 boards. 



In order to determine the ratio l>ctween the velocity o n the pro- 

 of the wheel and that of the water which drives it, Pa- per velocity 

 rent and Pitot considered only the action of the fluid up- und . CT '. 

 on one floatboard v and consequently they made the force 

 of impulsion proportional to the square of the relative 

 velocity, or to the square of the difference between the 

 velocity f the stream and that of the floattxjaril. Desa- 

 pnliers, Maclaurin, Lambert, At wood, Do Buat, and 

 l)r Robison, have gone noon the same principle, and 

 hnve therefore fallen into the same error, of making the 

 velocity of the wheel | of the velocity of the current 

 when the effect is a maximum. The Chevalier de 

 Borda, whose valuable Memoirs have been too much 

 overlooked by later writtr-, hi- hur\<r, correct- 

 ed this errror. He has shewn, that the supposi- 

 tion i* perfectly correct when the water impels a 

 single floatboard ; for as the number of particles 

 which strike the floatboard hi a given time, and also 

 tlie momentum of these, are each as tlie relative ve- 

 locity of the floatboards, the momentum must be as the 

 square of the relative velocity, that i, M == R'. M 

 being the momentum, anil R the relative velocity. Rut 

 M the water act* on more than one float board at once, 

 tlie number acted upon in a given time will be M the 

 velocity of the wheel, or inversely M the relative velo- 

 city ; for if we increase the relative velocity, the velo- 

 city of the water remaining the same, we must dimi- 

 nish the velocity of the wheel. Consequently, we shall 



}{ 

 have M == -g- or M == R; that is, the momentum of the 



water acting upon the wheel, varies M the relative ve- 

 locity. 



Now, let V be the velocity of the stream, F the 

 force with which it would strike the floatboard at rc<t, 

 and v the velocity of the wheel. Then the relative ve- 

 locity will be V r ; and since the velocity of the wu. 

 ter will be to its momentum, or the force with which 

 it would strike the floatboard at ret, as tlie relative ve- 

 locity is to the real force which the water exerts against 

 the moving floatboards, we shall have \ : V v = 



v. But the effect of the wheel 



is measured by the product of the momentum of the 

 water and the velocity of the wheel, consequently the 

 effect of the undershot wheel will be 



