OF HYDRAULICS. 



.(«N 



"JRNiA-.y 



63 



given quantity of a fluid acts upon a body, 

 its effect is simply as the relative velocity. 



For the length of the stream being given, the time of its 

 operation will be inversely as the relative velocity, and the 

 eflect being as the pressure and the time, or as the square 

 of the velocity directly, and inversely as the velocity, wriU be 

 simply as the velocity. If the fluid acts in such a manner 

 that the time does not vary, the proposition may be proved 

 from considering simply the quantity of motion that the 

 fluid loses, which must be the measure of the force with 

 which it acts on the solid. 



394. Theorem. The rotatory power of 

 a limited stream is greatest when it impels 

 an obstacle moving with half its own ve- 

 locity. 



For the rotatory power is as the force and the velocity 

 conjointly j now the length of the stream being limited, its 

 force is simply as the relative velocity, or as a—v, a being 

 the velocity of the stream, and v that of the obstacle ; but 

 av — «v is a maximum when oi'~2!)(', or 2ii~o. 



395. Theorem. When the surface of an 

 incompressible fluid, contained in a narrow 

 prismatic canal, is elevated or depressed a 

 little at any part above the general level, if 

 we suppose a point to move in the surface 

 each way with a velocity equal to that of a 

 heavy body falling tbrough half the depth 

 of the fluid, the surface of the fluid at the 

 part first affected will always be in a right 

 line between the two moveable points. 



The particles constituting any column of the fluid are 

 actuated by two forces, derived from the hydrostatic pres- 

 sures of the columns on each side, and these pressures are • 

 supposed to extend to the bottom of the canal, with an in- 

 ■ tensity regulated only by the height of the columns them- 

 selves; and this supposition would be either perfectly or 

 very nearly true if the particles of the fluid were infinitely 

 elastic, or absolutely incomjlressible. The difference of 

 these forces constituting a partial pressure, is the immediate 

 cause of the horizontal motion, and the vertical motion is the 

 effect : and this difference is every where to the weight of 

 the column, or of any of its portions, as the difference of the 

 heights to the thickness of the column, or as the fluxion of 

 the height to that of the horizontal length of the canal. 

 Such therefore is the force acting horizontally on any ele- 

 DMntary column ; but the elongation or abbreviation of the 



column depends on the difference of tbe velocities with 

 which its surfaces are made to advance, and this elevation 

 or depression is therefore to the whole height, as the varia- 

 tion of the fluxion of the length, produced by the operation 

 of the force, is to the whole fluxion of the length. While 

 therefore one of the supposed moveable points describes 3 

 given elementary arc, and the column is elevated or de- 

 pressed through its versed sine, which expresses half the 

 second fluxion of the height, its limits will approach each 

 other horizontally through a space as much less, as the 

 fluxion of the length is less than the whole height, and the 

 whole horizontal velocity being as much greater than this 

 relative velocity, as the force is greater than its fluxion, or 

 as the first fluxion of the height is greater than its second 

 fluxion, it follows that the whole horizontal velocity will 

 describe a space equal to half the first fluxion of tlic height, 

 diminished in the ratio of the fluxion of the length to the 

 height ; but if the force were altered so as to become equal 

 to that of gravity, or in the ratio of the fluxion of the height 

 to that of the length, the space described would become 

 equal to half the fluxion of the length, diminished in the 

 ratio of the fluxion of the length to the height ; and if the 

 time were increased in the ratio of the elementary arc, or 

 the fluxion of the length, to the height the space described 

 would be increased in the duplicate ratio, and would be- 

 come equal to half the height : since therefore the move- 

 able point describes a space equal to the depth, in the time 

 that half that space would be described by the action of 

 gravity, its velocity is equal to that which is acquired by a 

 heavy body in falling through half the depth, and the sur- 

 face of the fluid will initially describe a space equal to the 

 versed sine of an arc thus described by the moveable point. 

 In this manner the initial change of situation of every 

 part of the given surface may be determined, and the figure 

 which it^wiU have acquired at the end of any instant may 

 be considered as determining the acceleration of the motion 

 for a successive instant, which will always be such as to 

 add to the space described with the velocity acquired at the 

 beginning of the instant, a space equal to the mean of the 

 versed sines of the equal elementary arcs of the new curve 

 on each side of the point. But the sura of these versed.sincs 

 is always half the sum of the second fluxions of the height of 

 the original surface at equal distancesoneach side, correspond- 

 ing to the placeof the moveable points,.for the extremities of 

 the new elementary arcs being determined by the bisections 

 of two equal ch.ords, removed to the distance of the arc on 

 each side, the sagitta at each end is half of the excess of the 

 increment on one side above the increment adjoining to the 

 corresponding one on the other side, and the sum of the sa- 

 gittas is therefore halfof thesuraof tbe differences of the in- 



