sss

ittSCELLANEOUS

TRACT 3 Sr

AC . cb is equal the rectangle ae . eb : which makes be thesame for both. Or, if on the diameter ab a semicircle bedescribed ; then, because the squares of the ordinates ch,Di, ek are equal to the rectangles ac . cb, &c ; therefore thedistances bf, bg are as the ordinates ch, di. And hencealso it follows that the projection from the middle point i>will be farthest, for di is the greatest ordinate.

These are the proportions of the distances: but for theabsolute distances, it will be thus. The velocity throughany hole c, is such as will carry the water horizontallythrough a space equal to 2ac in rite time of falling throughAc: but, after quitting the hole, it describes a parabola, andcomes to f in the time a body will fall through cb ; and tofind this distance, since the times are as the roots of thespaces, therefore v"AC : ^/cb : : 2ac : 2f(AC . cb) = 2ch =bf, the space ranged on the horizontal plane. And thegreatest range bg = 2di, or 2ad, or equal to ab.

And as these ranges answer very exactly to the experi-ments, this confirms the theorjr, as to the velocity assigned.

PROBLEM XXIV.

To assign the Time of emptying a Vessel of Water, or otherFluid, through a Hole in the Bottom, as acbe.

Put h = ab the first height of the fluid

above the hole; !

= db the variable altitude at any Jtime;

a, = the area of the orifice b ;a' = the area of the descending sur-face CE.

Now the velocity of any issuing uniform-fluid, being thesame as that acquired by a body in falling through the height»b of the fluid, which is as the square-root of the height; and32 being the velocity acquired in falling through the space