LECTURE XXIIt.

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The effect of ajutages of different kinds, on the quantity of water dis-charged through an orifice of a given magnitude, may be most convenientlyexhibited by placing them side by side at the same height in a reservoir, andsuffering the water to begin to flow at the same moment through any two ofthem; the quantities discharged in a given time will then obviously indicatethe respective velocities. If a very long pipe were employed/some time wouldbe required before the velocity became uniform ;but in such cases the retardationarising from friction is so considerable, as to cause a still greater deviation fromthe quantity which would be discharged by a shorter pipe in the same time.

When the aperture, through which a fluid is discharged, instead of beingevery way of inconsiderable magnitude, is continued throughout the heightof the vessel, and is every where of equal breadth, the velocity must bematerially different at different parts of its height; but we may find thequantity of the discharge, by supposing the whole velocity equal to twothirds of the velocity at the lowest point. And we may find the quantitydischarged by an orifice not continued to the surface, hut still of consider-able height, by subtracting from the whole discharge of an orifice so con-tinued, that which would have been produced by such a portion of it, as musthe shut up, in order to form the orifice actually existing. But in this case,the result will seldom differ materially, from that which is found by consi-dering the pressure, on the whole orifice, as derived from the height of thefluid above its centre.

When a cylindrical vessel empties itself by a minute orifice, the velocity ofthe surface, which is always in the same proportion to the velocity of the fluidin flic orifice, is, therefore, uniformly retarded, and follows, in its descent thesame law as a heavy body, projected upwards, in its ascent; consequentlythe space actually described, in the whole time of descent, is equal to half ofthat which would have been described, if the initial motion had been un 1 'tormly continued; and in the time that such a vessel occupies, in emptyio»itself, twice the quantity of the fluid would be discharged if it were kepi-tull by a new supply. This may he easily shown, by filling two cylindricalvessels, having equal orifices in their bottoms, and while the one is left toempty itself, pouring into the other the contents of two other equal vesselsin succession, so as to keep it constantly full; for it will be seen that bothoperations will terminate at the same instant.