heights AB, A C.* Guillielmini sought for the cause of this augmentation in the weight of the atmosphere, and determined the velocity at C to be the same as would arise from the whole height A C. In his reasoning, he supposes that the pressure at C is the same for the state of motion as for that of rest; which is not true. In the experiments he made upon this object, he paid no regard, either to the diminution of expenditure produced by the irregularity of the inner surface of the tubes, nor the augmentation occasioned by the form of the tubes themselves. By a singular accidental concurrence, one of these errors compensated for the other. I know of no other decisive experiment on this head since Guillielmini. I shall, therefore, proceed to establish the proposition upon the principle of virtual ascension combined with the pressure of the atmosphere, and that in a manner which shall be clear of every objection, of theory as well as of experiment.

Let BLKO, Fig. 7, represent a conical tube adapted to the form of the contracted vein; the cylindrical tube LCQK is of the same diameter as the contracted part. The fluid stratum LK, continuing to descend through L C, tends to accelerate its motion, according to the laws of gravitation; and, consequently, when it passes from LK to MN, it tends to detach itself from the stratum which follows, or, in other words, it tends to produce a vacuum between L K and MN; and the same effect takes place through the whole length of the tube L C. The pressure of the atmosphere becomes active, as far as is necessary to prevent the vacuum; and its action is alike, both at the surface of the fluid at A, and at the inferior extremity of the tube at C. At A it increases the expenditure, and at Cit destroys

* Mouvement des Eaux, part iii. disc. 2.

+ Epist. Hydrostatic. Oper. tom. i. p. 212.

When I speak of the form of the contracted vein, I always mean to express the conoid formed by the fluid issuing from an orifice through a thin plate. V.

the sum of the accelerations which would be produced along L C, so that the fluid remains continuous in the tube.

Let T represent the time which the continuous column of fluid L CQ K employs to pass through the tube L C, whatever may be the velocity at L, and the successive acceleration from L to C. And, if we suppose this same column to return upwards from D to E, it will pass through the space D E = L C in the same time T, during which it will lose all the acceleration it acquired from L to C. The pressure of the column E D, continued for the time T, is, therefore, the quantity required to destroy the successive acceleration from L to C, and to prevent the fluid from ceasing to be continuous in the tube LC; consequently, that part of the pressure of the atmosphere, which is exerted at C Q to destroy the sum of the accelerations through L C, is equal to the pressure of a column ED of a fluid, homogeneous to that of the reservoir A B. And since the same pressure must also be exerted on the surface A of the reservoir, if we take FA = L C, the fluid at L K will possess the velocity which is proper to the height F L = A C; without considering the retardation which the internal inequalities of the tube L CQ K must produce.

EXPER. 9.-1. The orifice P (Fig. 1) through a thin plate is circular, and eighteen lines in diameter. The charge of fluid above the centre of the orifice is forty inches. Four cubic feet of water were emitted in thirty-eight seconds.

2. To the orifice P, Fig. 1, I applied the tube A C D, Fig.4, the upper end of which A C had the form of the contracted vein. The diameter at A was eighteen lines, its length, A D, thirtyone inches, and the situation of the tube horizontal. The expenditure of four cubical feet was made in forty-eight seconds.

3. The same orifice and the same tube were applied to the

horizontal bottom of the reservoir Fig. 7, so that the tube was vertical, and A C = forty inches, or the height of the charge in the two former experiments. The four cubic feet flowed out in forty-eight seconds, as in the second experiment.

EXPER. 10. The last described experiment was repeated with a circular aperture of 11.2 lines in diameter. The extremity AC of the tube Fig. 4, had the form of the contracted vein; the end A having the same diameter as that of the orifice. The other circumstances were as in the preceding cases. In the disposition, according to the first case, four cubical feet of water flowed out in ninety-eight seconds; in the second case, the time was 130 seconds; and in the third case, 129 seconds.

In each of these two experiments, the tubes and the expense of water were the same for the second and the third cases; whence it follows, that the force by which the expenditure was governed was the same in both cases. Now, the force which acts in the second case is the same as in the first; and, consequently, the same force likewise acts in the first and third cases. All the difference of the result, between the first case and the two following, arises from the retardation produced by the inequalities of the internal surface of the tubes.

EXPER. 11.-The height A B, Fig. 7, being constantly 32.5 inches, and orifice B O eighteen lines in diameter, the tube BO CQ was applied to the orifice itself, the superior extremity of this tube having the form of the contracted vein. When the length of the tube was varied, the times of the efflux of four cubic feet of water were as in the following table.

*

The fifth column of this table is calculated from the proportion of retardation produced by the irregularities of the internal surface of the tubes in the following experiment. Citizen Bossut has observed, that these retardations increase rather in a less ratio than the velocity of the stream. This is, perhaps, the reason of the difference observed between the fourth and fifth columns.

EXPER. 12.-I applied to the orifice P, Fig. 1, the same tubes as in the foregoing experiment, one after another, in a horizontal position, the height of the charge being constantly 32.5 inches above the centre of the orifice. The times of emission were as in the following table.

I must here observe, that the viscidity or mutual adhesion of the particles of the water is of very little consequence to the increase of expenditure through the orifice B O, Fig. 7, by the additional tube B C. For, as soon as a small hole is opened at K, the increase of expenditure diminishes or entirely ceases, and the fluid is no longer continuous in the tube.

We will now return to tubes in the horizontal and ascending positions.

PROPOSITION V.

In an additional conical Tube, the Pressure of the Atmosphere increases the Expenditure, in the Proportion of the exterior Section of the Tube to the Section of the contracted Vein, whatever may be the Position of the Tube, provided its internal Figure be adapted throughout to the lateral Communication of Motion.

We have seen (Proposition III.) that the pressure of the atmosphere increases the expenditure through additional tubes, whatever may be their position. We shall, in the next place, examine the mode of action by which the atmosphere produces this augmentation, and determine the result from its cause. I shall begin with the case best adapted to favour the action of the atmosphere, which is that of conical diverging tubes of a certain form, which we have not yet considered.

Let the extremity A B, Fig. 10, of the tube A BEF be applied to an orifice formed in a thin plate. The part A B CD is nearly of the figure of the contracted vein, which form has

Gravesande and others have attributed the increase of expenditure, through descending tubes, to the natural cohesion of the particles of water. V.