with a thin end, the cause just announced, joined to the diminution of the velocity, from the length of the tube being always very great when compared with its diameter, ought to produce a considerable diminution in the expenditure. M. Hachette verified this conjecture by means of a capillary tube whose length was 49.3 millimetres, and its diameter 1.19 millimetre. This tube terminated in a cone towards its extremity, occasioned under a pressure of 24 centimetres a diminution of 0.60 in the expenditure, calculated according to the theorem of Torricelli.

When we increase indefinitely the length of a capillary tube, we at last reach a limit beyond which the liquid flows out only drop by drop; but this limit varies with the height of the liquid above the orifice, as we shall see immediately.

Height of the Liquid above the Orifice.

The contraction of the vein diminishes with the height, or which is the same thing, with the pressure resulting from it. Thus, for example, while an orifice of 27 millimetres of diameter gives, under a pressure of 15 centimetres, a contraction of about 0.40; the same orifice, under a pressure of 16 millimetres, gives only a contraction of 0.31.

Since the fluid vein has a tendency to contract in proportion as the pressure increases, it was natural to think that when a pipe is employed, the fluid, by pressures always increasing, ought to tend more and more to detach itself from the inside of the pipe, and at last to separate itself altogether. This accordingly actually happens. The pressure necessary to produce the separation diminishes, as was to be expected, as the length of the pipe increases. It is less for a conical pipe than for a cylindrical one, and decreases at the same time as the angle at the summit of the cone, which is under consideration. M. Hachette found that for a pipe of six millimetres in length and 94 in diameter, it was still superior to 30 millimetres. He destroys, therefore, an opinion supported by Mr. Vince,* an English philosopher, that the flow cannot take place with the tube full in pipes shorter than six millimetres.

When the height of the liquid above the orifice becomes very small, the fluid vein at last acquires a particular form, very different from that which it had before, and which seems independent of the form of the orifice. M. Hachette calls this kind of veins secondary veins. He has observed them alike with orifices and pipes of all figures and sizes.

If we make the height of the liquid decrease indefinitely after having obtained secondary veins, we at last reach a limit beyond which the flow ceases to be uninterrupted. M. Hachette has par

two tubes were in the same plane, and the first rose above the second. Instead of diminishing the expenditure from 1 to, as Borda had done, I diminished it only in the ratio of 1 to 0.62.

* See his memoir, Phil. Trans. lxxxv. for 1795.

ticularly examined the laws of this last phenomenon in the case when cylindrical capillary tubes are employed as pipes. Six experiments made upon similar tubes of different lengths, and of the same diameter, appear to prove that the limit in question is proportional to the length of the tubes.

When the vessel containing the water has a very small size relative to the orifice, the form of the vein is sensibly altered, and becomes very irregular; but we can always make this irregularity disappear by increasing sufficiently the height of the liquid. †

Nature of the Fluid.

The experiments above related were made with water. Most of the phenomena remain the same when mercury is substituted for water. Thus, for example, the contraction relative to an orifice of one millimetre in diameter with thin sides, and that which under a pressure of 24 centimetres, a capillary tube of 49.3 millimetres in length, and 1.19 millimetre in diameter, gives, will be for mercury as for water, the first 0.31, and the second 0.60.

Alcohol, whose molecules adhere less to each other than those of water, flows out more readily. For the same reason, the pressure necessary to detach a fluid vein from the inside of a pipe is smaller for alcohol than for water.

When oil is substituted for water, the viscosity of the oil increases considerably the duration of the flow of the fluid through small ori

* These experiments were made upon a glass capillary tube, 0.53 millimetre in diameter. Having put it in a vertical position, water was made to flow from it by means of pressures measured exactly by means of the apparatus described in p. 219 of this report,

The length of the tube was at first 980 millimetres; and having successively diminished this length, five other tubes were obtained of the same diameter, and of the following lengths :

780, 580, 380, 180, 90, millimetres.

The constant flow ceased in each under the following pressures:·

586, 464, 342, 233, 120, 52 millimetres.

The pressures, calculated on the hypothesis that they are proportional to the lengths of the tube, would be

466, 346, 227, 107, 53, millimetres.

The small differences between the results of calculation and observation may be owing to a slight curvature in the tube, to the inequality of its interior sections, or to the uncertainty under which all these observations labour.

This experiment may be repeated with capillary tubes of different diameters, taking care that for water, for example, the diameters are below a millimetre; otherwise the thread would be constant, how small soever the height of the liquid above the inferior orifice of the tube.-H. C.

+ I avoided by the same means the helical motions in the capillary tubes which I used to study the motions of liquids in these tubes.-H. C.

This result agrees with the following experiment of M. Gay-Lussac, which M. de Laplace has related in his Mecanique Celeste, supplement to book x. p. 54.

A disc of glass of the diameter 118.366 millimetres was moistened successively with water and alcohol, and placed in contact with the surface of these liquids: the weights of the liquid column raised at the instant that it detaches itself from the disc are equal to 59'4 grammes and 31.147 grammes.-H. C.

fices. Through an orifice of one millimetre in diameter, the time of the flow of these two liquids was in the ratio of one to three.

The nature of the fluid is one of the principal causes on which depend the continuity or discontinuity of the jet in the flow through capillary tubes. When water was employed, the thread remained continuous at all pressures for a tube with a diameter equal to or greater than a millimetre. But when oil was used, the flow through a similar tube, whose length did not exceed five centimetres, was only drop by drop under a pressure of a column of oil more than a metre in height.

Surrounding Medium.

In experiments on the flow of a fluid by a given orifice or pipe, the surrounding air may influence in two ways: 1. By modifying the pressures on the orifice by the liquid under consideration. 2. By opposing a certain resistance to the emission of the liquid, or to its motion. That the first of these two effects may become sensible, it is necessary that the vertical pressure exercised from the top to the bottom on the upper surface of the liquid, and the pressure in a contrary direction on the exterior surface of the orifice or pipe should be very different from each other. This happens when we leave the upper part of the vessel containing the liquid exposed to the open air, and place the orifice or the pipe through which the liquid flows under the receiver of an air-pump, in which the air may be rarefied at pleasure. By means of this artifice, and by diminishing progressively the elastic force of the air under the receiver, we observe the same phenomena which are produced in the open by the gradual augmentation of the height of the liquid. We have even the advantage of being able to determine a very considerable pressure at little expense. It was by this method that M. Hachette was able to determine the diminution of the expenditure under a pressure equivalent to 10 metres of water, for capillary pipes terminated in cones towards the orifices-a diminution which was found the same as for pipes with thin sides and of a large diameter entirely plunged into a liquid.

air

If, instead of increasing the pressure, we wish to diminish it, it obviously would be sufficient to leave the given orifice or pipe exposed to the free air, and to put the upper surface of the liquid in contact with air rarefied under the receiver of an air-pump.

It remains for us to speak of the resistance opposed to the issue and to the motion of the fluid vein by the surrounding medium. Some philosophers have thought that we ought to ascribe to that resistance the changes of form which the vein experiences under variable pressures; but this conjecture is destroyed by the experiments of M. Hachette. He observed no difference in the form of the fluid veins produced by the flow of water and mercury through a triangular orifice in the air and in a vacuum.

The flow of a liquid through small cylindrical tubes seems entirely

to depend upon the resistance and density of the surrounding medium. Mr. Matthew Young* had already remarked that in this case, if we place the apparatus under the receiver of an air-pump, the flow continually decreases with the density of the air, and that in the open air the vein runs in a full stream filling the pipe, while in a vacuum it detaches itself from the sides of the pipe. But that philosopher does not appear to have suspected the difference which exists in this respect between tubes of a great and of a small diameter. M. Hachette has ascertained that a tube of 6.6 millimetres in diameter only gave two different products for all densities of the air, according as the fluid vein filled or did not fill the pipe. But when he employed a tube whose diameter was reduced to three millimetres, he obtained, like the British philosopher, an expenditure varying with the density of the air. Mr. Young concluded from his experiments that this expenditure reaches its maximum when the elastic force of the air is equivalent + to the weight of the liquid contained in the pipe, and that in this case the liquid fills the pipe; but this conclusion appears very doubtful. All that we can be sure of is, that for tubes of a very small diameter, when we diminish the elastic force of the air beyond a certain limit, the expenditure continually decreases. M. Hachette supposes with much probability that in that case the vein fills only a part of the pipe, and he ascribes this effect to the compression coming from the air, which endeavours to enter into the pipe to replace that which the motion of the liquid has necessarily carried off. When the diameter of the tube augments, a double current of air may be established, and the effect of which we are speaking ceases to take place.

It is evident from what has been said that M. Hachette has determined with much care the principal circumstances of the pheno→ mena which the motion of fluids presents, and sometimes even the laws of these phenomena. Still some questions relative to this subject remain to be resolved; as, for example, what ought to be the thickness of the walls of an orifice in order that it may exercise a marked influence on the expenditure? According to what law, when this influence is abstracted, does the contraction vary with the height of the liquid and the diameter of the orifice? Supposing the diameter given, what is the pressure at which the fluid vein changes into a secondary vein, and that at which the flow ceases to be constant? How does the pressure capable of separating a fluid vein from the sides of a cylindrical tube vary with the diameter, the length of the pipe, and the elastic force of the surrounding air? Finally, what length must we give to a cylindrical pipe of a deter

* See his papers, Memoirs of the Irish Academy, vol. vii.

+ I demonstrated in the first memoir on the flow of liquids through pipes, that when we increase the velocity of the liquid which issues from a pipe which it fills, the liquid vein detaches itself from the inside of the pipe, even when the elastic force of the medium in which the flow takes place is very superior to the weight of the liquid contained in the pipe.-(Note of M. Hachette.)

minate diameter to obtain the maximum of expense? These are so many problems which we will propose with confidence to M. Hachette. We think that, in engaging him to continue this kind of researches, the Academy ought to approve of his memoir, and order it to be printed in the Recueil des Savans Etrangers.*

ARTICLE X.

Proceedings of Philosophical Societies.

ROYAL ACADEMY OF SCIENCES.

Analysis of the Labours of the Royal Academy of Sciences of the Institute of France during the Year 1816.

PHYSICAL PART.-By M. le Chevalier Cuvier, Perpetual Secretary. (Continued from p. 146.)

BOTANY AND VEGETABLE PHYSICS.

One of the most important botanical considerations, and which connects it more than any other branch of natural history with the physical sciences in general, is vegetable geography, or the science of the laws of the distribution of plants according to the height of the pole, the elevation of the soil, the temperature, and the dryness or moisture of the climate.

M. de Humboldt, whose travels have advanced so remarkably this branch of knowledge, as well as several others, has just published a kind of complete treatise of it, under the title of Prolegomena de Distributione Geographica Plantarum secundum Cœli Temperiem et Altitudine Montium,t a work in which he gives at the same time profound researches on the distribution of heat, whether relative to the position of places, or to the seasons of the For not only the lines under which the mean annual temperature is the same are far from being parallel to the equator; but the places which have their whole mean heat equal are far from having their summers and winters similar. This mean heat may be more or less unequally spread through the whole of the year, and it is obvious

year.

In a third memoir I shall examine the motion of viscid liquids, I shall com. pare with each other the liquids of this kind which we obtain by dissolving in water gum, sugar, soap, glue, mucilage, &c. Bringing all these liquids to the same density, I shall measure the velocity of their flow, the difference of which will depend in that hypothesis on the adherence of the particles of liquid to each other, and to the sides of the vessel.

M. Petit and myself have ascertained, by an observation on the refraction, that when a liquid flows in a glass prism, taking care that the sides of the prism are not altered by the motion, the density of the liquid is the same, whether in a state of rest or motion.-H. C.

+ Paris, 1817, one volume, 8vo.

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