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and :: by similar A, we get
2XU F. (x − x)} ::
5: 2 the defective side of the rhombus; wherein is used according as the descending surface is > or < than half the rhombus. Hence this descending surface is a 6
- 1) ....... (a)
according as + or - is taken, the former value obtaining from a to r
and the latter from
a6 Now supposing the section where the descending sur
2 face first changes its form from the segment of a rhombus to an isosceles A to be an orifice, the time of the descent of the fluid to that section is (696) aov6
W x ao jo
2x} + C)
the time of the descent from x = a 10 x =
Again, the time of the fluid's descent in the pyrainidal portion of the vessel is (696 and (b))
it is 4u 7 6 ce
a 3x3 215x’AN 9 Hence the whole time required is
a2 , a T + T =
x 13-2/2 + 13}
X (S – 2).
716. In the first place let us seek the locus of the intersections of the parabolas described by the fluid issuing through the holes in any given vertical line of the surface.
Let y be the variable ordinate of any of these parabolas, x the corresponding ordinate measured from the top of the vessel, and not from the vertex or hole, and p the variable parameter of all the parabolas; then their general equation is (Vince, Hyd. prop. XXXIX.) y X
Now differentiating this equation on the supposition that p alone is variable, and putting the result = 0, and thence deducing p in terms of r, substitute for p in the equation, and the transformed equation will belong to the required locus. See Appendix to Vol. II. of Simpson's Flucions, pp. 339, &c., or pp. 47, 48, 67, 68, of Vol. II. of this work. We thus get for the locus the equation
which belongs to a straight line inclined to the horizon at an 2 of
45°, and meeting the top of the cylinder. Hence it follows, that the boundary of the water issuing from the cylinder is the surface of the frustum of a cone, the radii of the upper and lower ends being
r and a ta where r is the radius of the base of the cylinder, and a its altitude.
717. Let A be the area of any section parallel to the horizon, a that of the orifice, and h the altitude of the vessel ; then the time of emptying is (Vince's Flux. p. 268,)
Again, the time of falling through 2h is
= 2 g
9 ::t:t::A: a.
718. The area of any section parallel to the horizon is
nyo = ptx p being the parameter of the generating parabola, and the abscissa measured from the vertex. Also if A denote the area of the orifice, and h the whole height of the vessel ; then the velocity at the orifice is
2gh when the vessel is quite full; and the quantity discharged in each second is (by the question)
AV2gh = nQ Q being the given supply.
Now when the descending surface becomes stationary, equation (1) gives us
and the time of arriving at this position, equation (2) informs us is infinite, unless when n = 1, and then this time is zero. Hence we conclude that unless the influx is equal to the efflux at the very beginning of the discharge, the surface of the fluid will never be quiescent.
Although this conclusion is deduced after the methods expounded in all the Elementary Treatises (see Prony's Architecture Hydraulique, Tom. I. p. 342, and Bossut, Vol. I.; also Leybourne's
Mathematical Repository) yet it must be confessed it is far, from satisfactory. If the velocity of the descending surface be thus righììy estimated, it may easily be shewn that the expression for the time of reaching the state of quiescence will always involve log. (o), or be co, whatever may be the form of the vessel; in other words, no such state of quiescence is possible. This seeming contrary to experience, we intend prosecuting the subject somewhat farther at our leisure, and shall probably insert the results in some Scientific Journal.
719. Let y be the radius of the sphere, x the altitude of the fluid at any epoch, and A the area of the orifice; then the velocity at the orifice being
w 2gx that of the descending surface is A
√x ✓ 292 =
whence the position of the surface when its velocity is a minimum is obtained, and this minimum velocity is
2A N 39
Although this velocity is a minimum, yet it is not the least velocity of all, that at the orifice being 0.
The reader must observe, that at the orifice the descending surface is = orifice; for otherwise it might appear from equation (1) that this is ao instead of o.