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720. Let A be the area of the orifice of the cylinder, h its 'altitude, and r the radius of its base; then if the water descend through h — x during the time t, the moving force at the end of
that time which causes pressure and an increase of velocity at the orifice is
Hence the force which actually accelerates each particle of the fluid downwards through the orifice is
Hence the velocity at the orifice is due to the altitude
and which being taken between x = ō and x = h gives
Again, the force which accelerates the descent of Pis
where ds is the element of space described by P in the time dt.
Hence (see eq. a)
for the space described by P during the discharge of the water.
Let be the given inclination of the tube to the horizon, a the altitude of the orifice of the vessel, the observed distance at which the water strikes the plane from the vertical line passing through the orifice; then since the water describes a parabola, whose equation is (see p. 253,)
where y and x are the vertical and horizontal co-ordinates originating at the orifice, and v the velocity of the issuing fluid.
But if h be the required altitude of the fluid above the orifice, we have
Let p be the parameter of the generating parabola, the radius of the base of the paraboloid, y that of the required orifice, a the given altitude of the water above the vertex, and S the given space descended through in the given time T. Then at any variable altitude c y2 above the orifice, the velocity at the
which being reduced and resolved, will give the required
value of y.
Let B be the required altitude of the hole, a the altitude of the cone; then since the equation to the parabola described is (see p. 253,)
Hence by the solution of a quadratic equation, £ and .. the re
quired position may be found.
Let a be
variable altitude of the water, A the
area of the orifice, a the length of the axis, and ẞ the extreme ordinate; then since the area of the descending surface is
an integral which can be taken only between certain limits, as 0, and x = co. See l'hewell's Dynamics, p. 15.
Let 2a be the altitude of the water when it is in equilibro, 2b the distance between the axes of the two legs; then the centre of gravity of the water in the legs in the state of equi
librium is the middle point of the line joining the bisections of the water in the legs. Make this point the origin of the vertical and horizontal co-ordinates y, x; then supposing the water to have descended down one of the legs, it will rise as much in the other, and the line which joins the bisections of the axes of the fluid in the legs, being divided in the inverse proportion of the altitudes of the water, will give the centre of gravity corresponding to this position. Hence if m denote the distance to which the centre of gravity of the water has been depressed in one leg and elevated in the other, it is easily shewn that
or the required curve is the common parabola, whose principal parameter is ¿3.
Let y pa" be assumed as the equation to the required parabola. Also let a be the common altitude of the paraboloid and cylinder.